library(vegan)
## Lade nötiges Paket: permute
## Lade nötiges Paket: lattice
## This is vegan 2.6-4
library(shape) # plots nice arrows
setwd("Z:/COURSES/datenanalyse/vu_datenanalyse/unit_3") #set working directory
mara.raw<-read.table(file="data/MaraRiver.txt",header=TRUE) # water chemistry in 54 streams, 3 types of land use
dim(mara.raw)
## [1] 54 49

head(mara.raw)
##   site sampling.time diff.solar.noon landuse order     Q canopy Temp  cond   pH
## 1 Aa20         10:04          145.92       A     2 3.572    0.2 13.4 229.1 7.45
## 2 Aa21         14:01           91.08       A     1 6.600    0.2 17.0 205.4 8.03
## 3 Aa22         16:35          245.08       A     1 1.650    0.3 14.5 205.7 7.88
## 4 Aa23         10:05          145.22       A     1 5.472    0.4 12.1 181.8 7.96
## 5 Aa6b         17:03          270.78       A     2 1.800    0.2 17.5 404.4 7.79
## 6  Af5         14:04           94.38       F     3 4.200    0.4 14.5 129.4 7.94
##     turb  Alk       TSS      PO4         NH4      NO2         NO3       Br
## 1   0.02 2160  12.55102 5.16e-07 0.000022800 8.91e-07 0.000054500 2.45e-06
## 2  35.50 2160  59.44444 8.32e-07 0.000023600 6.08e-07 0.000007810 2.06e-06
## 3   0.02 2160  10.57143 1.37e-07 0.000027800 1.13e-06 0.000047300 1.39e-06
## 4   0.02 2160  26.96629 1.00e-06 0.000020700 2.17e-06 0.000124766 1.24e-06
## 5 388.00 3740 209.41176 1.37e-07 0.000446205 1.98e-05 0.000092700 2.47e-06
## 6  28.80 2160  10.21053 1.37e-07 0.000019300 6.95e-07 0.000034900 8.64e-07
##            Cl       Fl      SO4          Na           K          Ca          Mg
## 1 0.000265196 4.47e-05 8.58e-05 0.000930796 0.000423884 0.000328085 0.000152685
## 2 0.000237582 3.91e-05 5.09e-05 0.000843062 0.000323802 0.000323170 0.000146390
## 3 0.000220969 2.87e-05 3.56e-05 0.000622227 0.000373881 0.000372199 0.000157663
## 4 0.000215891 3.03e-05 4.66e-05 0.000716442 0.000299606 0.000300339 0.000132606
## 5 0.000354103 6.12e-05 5.97e-05 0.001070465 0.000989155 0.000507910 0.000253652
## 6 0.000124757 2.87e-05 5.13e-05          NA          NA          NA          NA
##        TDN  SUVA254      slope slope_short  SR_Helms E2.to.E3 E4.to.E6      FIX
## 1 7.82e-05 2.486708 0.01484335  0.01508883 0.8385724 5.729730 7.333333 1.432674
## 2 3.20e-05 2.749288 0.01304546  0.01313918 0.8853058 4.643519 5.000000 1.352617
## 3 7.62e-05 2.957983 0.01374024  0.01376633 0.8516180 4.986425 8.000000 1.371267
## 4 1.48e-04 3.069472 0.01360719  0.01317551 0.8079401 4.892857 6.714286 1.349814
## 5 5.59e-04 1.722983 0.01444961  0.01542405 0.8943684 5.288889 7.833333 1.481364
## 6 5.49e-05 3.088685 0.01376586  0.01382645 0.8333595 4.981043 6.428571 1.336503
##       FIX2       HIX beta.alpha      X7c_c1    X7c_c2    X7c_c3    X7c_c4
## 1 1.781212 0.8178391  0.7186987 0.002585346 0.5892004 0.4082030 0.2100508
## 2 1.709486 0.7345711  0.6978052 0.075614095 0.7089239 0.4878008 0.2858983
## 3 1.726269 0.7851819  0.6829999 0.037594156 0.8459288 0.5640788 0.3114274
## 4 1.701107 0.8066407  0.6676339 0.010937179 0.6222450 0.4279429 0.2482675
## 5 1.961382 0.6234517  0.7639094 0.222123376 1.8301573 1.7719178 0.5627895
## 6 1.722147 0.8536532  0.6777914 0.035057086 1.3009351 1.0052846 0.5806384
##       X7c_c5    X7c_c6    X7c_c7 humF.to.protF humF.per.DOC protF.per.DOC
## 1 0.03701974 0.3347390 0.1175206      9.815030    0.3153769    0.03213204
## 2 0.09037746 0.3795044 0.3062165      3.943447    0.2652603    0.06726611
## 3 0.05079516 0.4395259 0.2434638      6.511800    0.3026556    0.04647802
## 4 0.04149694 0.3223648 0.1405717      8.397779    0.2706781    0.03223210
## 5 0.63429053 1.1599349 1.5369801      2.224790    0.4052359    0.18214567
## 6 0.00000000 0.7400336 0.2134428     14.595144    0.5545706    0.03799693
##    DIC.d13C     pCO2    EpCO2
## 1 -2.673634       NA       NA
## 2 -8.590664 1184.457 3.060612
## 3 -4.565283 1753.501 4.531012
## 4 -7.214370 1094.980 2.829405
## 5 -4.717236 3730.961 9.640726
## 6 -9.946630       NA       NA
mara<-mara.raw[,c(4,9:26)] #only select columns of interest

which(is.na(mara), arr.ind=TRUE) #find rows with NAs
##      row col
## [1,]   6  15
## [2,]   6  16
## [3,]   6  17
## [4,]   6  18
mara<-mara[-6,] #delete row with NAs

landuse<-mara$landuse
wc<-mara[,-1]

apply(wc, 2, hist) #check normal distribution

pca<-prcomp(scale(wc),center=F,scale.=F)
pca<-prcomp(wc,center=T,scale.=T)   # equivalent to line above

summary(pca)
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Standard deviation     2.8392 1.8644 1.22808 1.10689 0.84577 0.80664 0.78730
## Proportion of Variance 0.4478 0.1931 0.08379 0.06807 0.03974 0.03615 0.03444
## Cumulative Proportion  0.4478 0.6410 0.72474 0.79281 0.83255 0.86870 0.90314
##                            PC8    PC9    PC10    PC11   PC12    PC13    PC14
## Standard deviation     0.66233 0.6392 0.53771 0.40863 0.3445 0.32772 0.29622
## Proportion of Variance 0.02437 0.0227 0.01606 0.00928 0.0066 0.00597 0.00487
## Cumulative Proportion  0.92751 0.9502 0.96627 0.97554 0.9821 0.98810 0.99298
##                           PC15    PC16   PC17    PC18
## Standard deviation     0.25252 0.21425 0.1038 0.07711
## Proportion of Variance 0.00354 0.00255 0.0006 0.00033
## Cumulative Proportion  0.99652 0.99907 0.9997 1.00000

pca$sdev # stdevs of PCs (squares are eigenvalues); eigenvalues correspond to the variance of their respective principal component (PC)
##  [1] 2.83921334 1.86442507 1.22808376 1.10689311 0.84576999 0.80663510
##  [7] 0.78729929 0.66233089 0.63915683 0.53770934 0.40862593 0.34454651
## [13] 0.32772103 0.29622209 0.25251770 0.21424557 0.10379779 0.07711088
eigenvals(pca)==pca$sdev^2
##  PC1  PC2  PC3  PC4  PC5  PC6  PC7  PC8  PC9 PC10 PC11 PC12 PC13 PC14 PC15 PC16 
## TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 
## PC17 PC18 
## TRUE TRUE

head(scores<-pca$x) # site scores on all PCs
##         PC1      PC2         PC3        PC4        PC5        PC6         PC7
## 1 -4.749326 3.557587  0.52830345  1.6885404  0.4637009  0.6815660  0.06534355
## 2 -3.830547 3.155164  0.07572526 -2.0332541  0.2644377  1.1129832 -0.55448962
## 3 -3.267373 4.108748 -0.32897368  0.7711051 -1.0759520 -0.2527134  0.44879703
## 4 -3.680797 2.292490  0.80951314  0.2051344 -0.2780356 -0.6345223  0.05431164
## 5 -8.999719 1.602003 -3.03923365 -1.2132318 -1.0636252 -1.2871947  0.42462524
## 7  2.315295 1.447056  0.53769126  0.3896117  0.5148749 -1.6163461 -0.06189673
##          PC8        PC9        PC10       PC11         PC12        PC13
## 1  0.9056796 -0.1326339 -0.36156648  0.3464919 -0.343851835 -0.09665447
## 2  1.2660092 -0.0486270 -1.04265251 -0.2685484 -0.237503649 -0.68256243
## 3 -0.1695538 -0.4501332 -0.45996058  0.5662810  0.124965534 -0.53160488
## 4 -0.5018929 -0.9758938 -0.49191459  0.1223105 -0.230524069  0.50704177
## 5  0.3163298 -0.1965387 -0.05913828 -0.1130011  1.000404068  0.11682215
## 7 -1.1804198 -1.0651942 -0.46082573 -1.2090894 -0.008212293 -0.32546687
##          PC14       PC15         PC16        PC17        PC18
## 1  0.36104029  0.1178840  0.004293307  0.04989172  0.05965282
## 2 -0.14602705 -0.1922231  0.155271851  0.28350624 -0.09569322
## 3 -0.40743689  0.4087700  0.288081091 -0.24071374  0.04519902
## 4 -0.01593147 -0.3959628 -0.098755194 -0.01316164  0.08656537
## 5 -0.17548689  0.1746190 -0.311008889  0.19590434 -0.02083158
## 7  0.58480671 -0.4687339 -0.016056662 -0.03962186  0.01500699

head(scale(wc) %*% pca$rotation) # to manually compute scores from variables and loadings
##         PC1      PC2         PC3        PC4        PC5        PC6         PC7
## 1 -4.749326 3.557587  0.52830345  1.6885404  0.4637009  0.6815660  0.06534355
## 2 -3.830547 3.155164  0.07572526 -2.0332541  0.2644377  1.1129832 -0.55448962
## 3 -3.267373 4.108748 -0.32897368  0.7711051 -1.0759520 -0.2527134  0.44879703
## 4 -3.680797 2.292490  0.80951314  0.2051344 -0.2780356 -0.6345223  0.05431164
## 5 -8.999719 1.602003 -3.03923365 -1.2132318 -1.0636252 -1.2871947  0.42462524
## 7  2.315295 1.447056  0.53769126  0.3896117  0.5148749 -1.6163461 -0.06189673
##          PC8        PC9        PC10       PC11         PC12        PC13
## 1  0.9056796 -0.1326339 -0.36156648  0.3464919 -0.343851835 -0.09665447
## 2  1.2660092 -0.0486270 -1.04265251 -0.2685484 -0.237503649 -0.68256243
## 3 -0.1695538 -0.4501332 -0.45996058  0.5662810  0.124965534 -0.53160488
## 4 -0.5018929 -0.9758938 -0.49191459  0.1223105 -0.230524069  0.50704177
## 5  0.3163298 -0.1965387 -0.05913828 -0.1130011  1.000404068  0.11682215
## 7 -1.1804198 -1.0651942 -0.46082573 -1.2090894 -0.008212293 -0.32546687
##          PC14       PC15         PC16        PC17        PC18
## 1  0.36104029  0.1178840  0.004293307  0.04989172  0.05965282
## 2 -0.14602705 -0.1922231  0.155271851  0.28350624 -0.09569322
## 3 -0.40743689  0.4087700  0.288081091 -0.24071374  0.04519902
## 4 -0.01593147 -0.3959628 -0.098755194 -0.01316164  0.08656537
## 5 -0.17548689  0.1746190 -0.311008889  0.19590434 -0.02083158
## 7  0.58480671 -0.4687339 -0.016056662 -0.03962186  0.01500699
head(loadings<-pca$rotation) # the variable loadings
##              PC1        PC2        PC3         PC4         PC5         PC6
## cond -0.26935347  0.1438792 -0.2841620 -0.03056776  0.12191093  0.08071683
## pH    0.04777530  0.1215468  0.3562802 -0.59539396 -0.49268822 -0.29216145
## turb -0.14155181 -0.3894707 -0.2177088 -0.32746528 -0.03611414  0.24629283
## Alk  -0.17098754  0.3503345 -0.1962747  0.09297582  0.10935101 -0.12209184
## TSS  -0.14508786 -0.3207217 -0.2732119 -0.44573924 -0.03791156  0.26821922
## PO4  -0.01446834 -0.2443586  0.4595895 -0.23241378  0.72222431 -0.10638692
##              PC7         PC8         PC9         PC10        PC11         PC12
## cond -0.41973560 -0.17562478 -0.09881877 -0.006566902  0.64824153 -0.064479508
## pH   -0.22576487 -0.26192817  0.06281913 -0.192775926  0.05253252 -0.003739136
## turb -0.04788218  0.07002402 -0.01614397  0.011308932 -0.32201794 -0.137669971
## Alk  -0.42805174 -0.15744036 -0.37748332 -0.112749767 -0.59639437  0.187029632
## TSS   0.02529992  0.16003510 -0.08747407 -0.150755304 -0.03128159  0.262605089
## PO4  -0.05401838 -0.17326809 -0.05875984 -0.015299767 -0.07294578 -0.177318022
##             PC13        PC14        PC15         PC16          PC17
## cond -0.12739932  0.02559159 -0.11856620 -0.336469042 -0.0003470607
## pH   -0.04129351  0.07363790  0.02360840  0.043866907 -0.0064374198
## turb -0.59531621 -0.04723956  0.24918988 -0.223721709 -0.1124305273
## Alk   0.01598820  0.05220398 -0.07854688  0.058152265  0.0221005482
## TSS   0.55485685 -0.08756056 -0.29311696  0.002605816  0.0204648859
## PO4   0.03082657  0.13171082 -0.19384168 -0.090576799  0.0664719576
##              PC18
## cond  0.135379479
## pH    0.017462207
## turb  0.005186753
## Alk   0.084188983
## TSS   0.030558062
## PO4  -0.005485131


# how many axes should be kept?
#there is not the one well-accepted method
#1. visual examination
#looking for a point at which the proportion of variance explained by each subsequent principal component drops off (elbow of scree plot)
screeplot(pca, npcs=length(pca$sdev), type = "lines") # plots eigenvalues vs. principal components (PCs) #
#2 Kaiser-Guttman criterion
# retain PCs with eigenvalues>1, PCs with eigenvalues<1 contain less informatin than one original variable
abline(h=1,col="red")

# eigenvalue of PC5<1
#3. variance explained method: set an "arbitrary" threshold of explained variance (e.g. 70%, 80%, 90%) and choose as many PCs till threshold is reached.
summary(pca)
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Standard deviation     2.8392 1.8644 1.22808 1.10689 0.84577 0.80664 0.78730
## Proportion of Variance 0.4478 0.1931 0.08379 0.06807 0.03974 0.03615 0.03444
## Cumulative Proportion  0.4478 0.6410 0.72474 0.79281 0.83255 0.86870 0.90314
##                            PC8    PC9    PC10    PC11   PC12    PC13    PC14
## Standard deviation     0.66233 0.6392 0.53771 0.40863 0.3445 0.32772 0.29622
## Proportion of Variance 0.02437 0.0227 0.01606 0.00928 0.0066 0.00597 0.00487
## Cumulative Proportion  0.92751 0.9502 0.96627 0.97554 0.9821 0.98810 0.99298
##                           PC15    PC16   PC17    PC18
## Standard deviation     0.25252 0.21425 0.1038 0.07711
## Proportion of Variance 0.00354 0.00255 0.0006 0.00033
## Cumulative Proportion  0.99652 0.99907 0.9997 1.00000

# -> the first 3-4 PCs seem useful, and just PC1 and PC2 alone are already explaining a lot of overall variance


# for a DISTANCE BIPLOT (focus is on sites, "scaling 1")
# each principal component has variance given by eigenvalue, loadings remain unscaled
str(landuse)
##  chr [1:53] "A" "A" "A" "A" "A" "M" "F" "M" "M" "M" "F" "A" "A" "A" "A" "A" ...
landuse<-as.factor(landuse)
plot(scores[,1:2],asp=1,pch=21, cex=2, bg=landuse) # note asp=1
arrows<-loadings*15 # with extension factor
Arrows(x0=0,y0=0,x1=arrows[,1],y1=arrows[,2],col="darkgreen")
text(x=arrows[,1]*1.2,y=arrows[,2]*1.2,labels=names(wc),cex=0.8)
#text(scores[,1:2], rownames(wc), pos=2)
legend("bottomright", legend=c("Agriculture", "Mixed", "Forest"), pch=21, pt.cex=2, pt.bg=c("black","green","darkred"), cex=1.5)

biplot(pca,scale=0)

# in this plot:
# 1) distances among sites are approximating true Euclidean distances in multivariate space
# 2) angles between arrows do not reflect correlations among variables
# 3) projecting site on descriptor at right angle gives its appr. descriptor value

# for a CORRELATION BIPLOT (focus is on variables, "scaling 2")
# each principal component is weighted by 1/sqrt(eigenvalue), so it has variance 1
var(scores[,1]/pca$sdev[1]) # just demo
## [1] 1
plot(scores[,1]/pca$sdev[1],scores[,2]/pca$sdev[2],pch=21,bg=landuse,asp=1)
# loadings are weighted by sqrt(eigenvalues)
arrows<-loadings*matrix(pca$sdev,nrow=nrow(loadings),ncol=ncol(loadings),byrow=TRUE)
arrows<-arrows*2 # choose extension factor
Arrows(x0=0,y0=0,x1=arrows[,1],y1=arrows[,2],col="purple")
# as alternative just compute correlation of scores with original data ("structure coefficients")
(structure<-cor(wc,scores))
##              PC1         PC2         PC3         PC4          PC5          PC6
## cond -0.76475195  0.26825195 -0.34897469 -0.03383524  0.103108608  0.065109031
## pH    0.13564426  0.22661496  0.43754197 -0.65903747 -0.416700907 -0.235667684
## turb -0.40189578 -0.72613887 -0.26736468 -0.36246906 -0.030544259  0.198668445
## Alk  -0.48547011  0.65317251 -0.24104174  0.10291430  0.092485802 -0.098483565
## TSS  -0.41193538 -0.59796155 -0.33552705 -0.49338569 -0.032064459  0.216355039
## PO4  -0.04107869 -0.45558830  0.56441437 -0.25725721  0.610835646 -0.085815422
## NH4  -0.59306916  0.02901967 -0.52211239 -0.05077205  0.146005150 -0.407246534
## NO2  -0.60159868 -0.44979709 -0.11708309 -0.06542819 -0.083433731 -0.478631077
## NO3  -0.51894312 -0.68971906  0.17118492  0.35407824 -0.145546662 -0.008334060
## Br   -0.92514493 -0.10433460  0.06301001  0.05315603  0.008796639  0.221187748
## Cl   -0.82983473 -0.12663016  0.38234130  0.16849740  0.028517422 -0.093976441
## Fl   -0.85617616 -0.08722719  0.14001888 -0.03859390  0.108893567 -0.027198849
## SO4  -0.81130553  0.18629947  0.30840708  0.17786031 -0.164712321  0.009860507
## Na   -0.88109521  0.13206079  0.10468309  0.04519175 -0.161228683  0.142295300
## K    -0.79637284  0.37181544  0.19278724 -0.11851266  0.026616893  0.075563024
## Ca   -0.73330911  0.56721172 -0.04684159 -0.17535185  0.084387922  0.046307940
## Mg   -0.76241145  0.49212149  0.03837621 -0.19104400  0.036005859  0.052768803
## TDN  -0.66409999 -0.60500271 -0.04421175  0.24134387 -0.151522130 -0.055259365
##              PC7         PC8          PC9         PC10         PC11
## cond -0.33045754 -0.11632172 -0.063160692 -0.003531084  0.264888300
## pH   -0.17774452 -0.17348312  0.040151278 -0.103657416  0.021466149
## turb -0.03769761  0.04637907 -0.010318528  0.006080919 -0.131584882
## Alk  -0.33700483 -0.10427761 -0.241271045 -0.060626603 -0.243702206
## TSS   0.01991861  0.10599619 -0.055909646 -0.081062536 -0.012782470
## PO4  -0.04252863 -0.11476081 -0.037556751 -0.008226828 -0.029807536
## NH4   0.14949569 -0.04291440  0.365863964 -0.155284000 -0.033708442
## NO2   0.07756756  0.19275231 -0.298888832  0.209087387  0.052794089
## NO3   0.01371875 -0.25604120 -0.048509024 -0.033794332 -0.013701263
## Br    0.04173239  0.00127664 -0.052263599 -0.162136464  0.069169573
## Cl   -0.03699389  0.13621525  0.008198462 -0.180611063  0.032692795
## Fl   -0.33251999  0.16703916  0.191260615  0.124728010  0.003445899
## SO4   0.05600956  0.30157467 -0.015244843 -0.159709162 -0.046927081
## Na   -0.13133773 -0.01459469  0.187683610  0.211586965 -0.073979848
## K     0.21133725 -0.08141740  0.106111667  0.223653004 -0.011006163
## Ca    0.21377568 -0.09313882 -0.149458288 -0.054964294  0.017660115
## Mg    0.32651015 -0.06474423 -0.077876661  0.034731711 -0.011551626
## TDN   0.04747067 -0.30977212 -0.004718641  0.007756368 -0.019322840
##              PC12         PC13          PC14         PC15          PC16
## cond -0.022216190 -0.041751438  0.0075807958 -0.029940066 -0.0720870031
## pH   -0.001288306 -0.013532752  0.0218131727  0.005961538  0.0093982907
## turb -0.047433709 -0.195097641 -0.0139934007  0.062924857 -0.0479313860
## Alk   0.064440408  0.005239670  0.0154639711 -0.019834478  0.0124588654
## TSS   0.090479668  0.181838259 -0.0259373713 -0.074017223  0.0005582845
## PO4  -0.061094306  0.010102514  0.0390156553 -0.048948456 -0.0194056782
## NH4  -0.019558805 -0.008153826  0.0261212836 -0.022615617  0.0068037616
## NO2  -0.018137139 -0.022027853  0.0217333801 -0.022298082  0.0303642603
## NO3  -0.015197048  0.066748065  0.0090589510  0.018038656 -0.0081950848
## Br    0.003987132 -0.074486267  0.1268513291 -0.005079333  0.1418328916
## Cl    0.108916049 -0.077892719 -0.1965043400 -0.029094888  0.0066897381
## Fl    0.021179746  0.098884906  0.0044583717  0.154701308  0.0292928111
## SO4  -0.064893733  0.034208880  0.1163038494 -0.018049823 -0.1085844000
## Na   -0.153097268  0.003712544 -0.0563765190 -0.126996360  0.0355530451
## K     0.216803483 -0.052579706  0.0705418244 -0.028508949 -0.0372784074
## Ca   -0.081684815  0.049612428 -0.0519750553  0.054216175 -0.0041896038
## Mg   -0.068998656  0.024034873 -0.0651154780  0.058400432  0.0043412431
## TDN   0.023214045  0.028784640 -0.0001785383  0.029159768 -0.0353732943
##               PC17          PC18
## cond -3.602414e-05  0.0104392312
## pH   -6.681899e-04  0.0013465262
## turb -1.167004e-02  0.0003999551
## Alk   2.293988e-03  0.0064918868
## TSS   2.124210e-03  0.0023563591
## PO4   6.899642e-03 -0.0004229633
## NH4  -9.032968e-03  0.0036488485
## NO2  -2.342515e-03 -0.0016575665
## NO3  -6.145284e-02  0.0262899055
## Br    8.706134e-03 -0.0017381812
## Cl    6.151965e-04 -0.0017187344
## Fl    6.209194e-04 -0.0016118828
## SO4   6.993261e-03 -0.0001298039
## Na   -2.090597e-04 -0.0059058664
## K    -1.630078e-02 -0.0026053297
## Ca   -3.672943e-02 -0.0461913573
## Mg    3.246239e-02  0.0471881766
## TDN   6.262864e-02 -0.0258731171
structure<-2*structure
Arrows(x0=0,y0=0,x1=structure[,1],y1=structure[,2],col="red")
text(x=arrows[,1]*1.3,y=arrows[,2]*1.2,labels=names(wc),cex=0.7)

biplot(pca,scale=1)

# in this plot
# 1) distances among sites are not approximating true Euclidean distances in multivariate space
# 2) angles between arrows reflect correlations among variables (NOT proximity of arrow heads)
# 3) projecting site on descriptor at right angle gives its appr. descriptor value

# PCA using the rda function from the vegan package
pca2<-rda(X=wc,scale=TRUE)
summary(pca2,scaling=1)
## 
## Call:
## rda(X = wc, scale = TRUE) 
## 
## Partitioning of correlations:
##               Inertia Proportion
## Total              18          1
## Unconstrained      18          1
## 
## Eigenvalues, and their contribution to the correlations 
## 
## Importance of components:
##                          PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Eigenvalue            8.0611 3.4761 1.50819 1.22521 0.71533 0.65066 0.61984
## Proportion Explained  0.4478 0.1931 0.08379 0.06807 0.03974 0.03615 0.03444
## Cumulative Proportion 0.4478 0.6410 0.72474 0.79281 0.83255 0.86870 0.90314
##                           PC8    PC9    PC10     PC11     PC12     PC13
## Eigenvalue            0.43868 0.4085 0.28913 0.166975 0.118712 0.107401
## Proportion Explained  0.02437 0.0227 0.01606 0.009276 0.006595 0.005967
## Cumulative Proportion 0.92751 0.9502 0.96627 0.975542 0.982137 0.988104
##                           PC14     PC15    PC16      PC17      PC18
## Eigenvalue            0.087748 0.063765 0.04590 0.0107740 0.0059461
## Proportion Explained  0.004875 0.003543 0.00255 0.0005986 0.0003303
## Cumulative Proportion 0.992979 0.996521 0.99907 0.9996697 1.0000000
## 
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores:  5.531195 
## 
## 
## Species scores
## 
##           PC1      PC2     PC3     PC4      PC5      PC6
## cond -1.48985  0.79582 -1.5718 -0.1691  0.67431  0.44646
## pH    0.26425  0.67230  1.9707 -3.2932 -2.72515 -1.61600
## turb -0.78295 -2.15424 -1.2042 -1.8113 -0.19975  1.36229
## Alk  -0.94577  1.93777 -1.0856  0.5143  0.60484 -0.67531
## TSS  -0.80251 -1.77397 -1.5112 -2.4655 -0.20970  1.48357
## PO4  -0.08003 -1.35160  2.5421 -1.2855  3.99476 -0.58845
## NH4  -1.15538  0.08609 -2.3516 -0.2537  0.95485 -2.79254
## NO2  -1.17200 -1.33441 -0.5273 -0.3269 -0.54564 -3.28203
## NO3  -1.01098 -2.04619  0.7710  1.7693 -0.95185 -0.05715
## Br   -1.80232 -0.30953  0.2838  0.2656  0.05753  1.51671
## Cl   -1.61664 -0.37567  1.7220  0.8420  0.18650 -0.64441
## Fl   -1.66795 -0.25878  0.6306 -0.1929  0.71215 -0.18651
## SO4  -1.58054  0.55270  1.3890  0.8888 -1.07719  0.06761
## Na   -1.71650  0.39179  0.4715  0.2258 -1.05441  0.97574
## K    -1.55145  1.10307  0.8683 -0.5922  0.17407  0.51814
## Ca   -1.42859  1.68275 -0.2110 -0.8762  0.55188  0.31754
## Mg   -1.48529  1.45998  0.1728 -0.9547  0.23547  0.36184
## TDN  -1.29376 -1.79486 -0.1991  1.2060 -0.99093 -0.37892
## 
## 
## Site scores (weighted sums of species scores)
## 
##          PC1      PC2       PC3      PC4       PC5       PC6
## 1  -0.858644  0.64319  0.095513  0.30528  0.083834  0.123222
## 2  -0.692535  0.57043  0.013691 -0.36760  0.047808  0.201219
## 3  -0.590717  0.74283 -0.059476  0.13941 -0.194524 -0.045689
## 4  -0.665461  0.41447  0.146354  0.03709 -0.050267 -0.114717
## 5  -1.627084  0.28963 -0.549471 -0.21934 -0.192296 -0.232715
## 7   0.418588  0.26162  0.097211  0.07044  0.093086 -0.292224
## 8   0.990156  0.31409 -0.773129 -0.34073 -0.067209  0.150470
## 9  -0.298376  0.20548 -0.191656  0.70420 -0.194790  0.113638
## 10 -0.509697  0.18895  0.022804 -0.32743  0.211013  0.044924
## 11  0.158309  0.31689 -0.173319  0.16996  0.237407  0.066935
## 12  0.764213  0.20918 -0.027062  0.06540 -0.147885 -0.096221
## 13 -0.178540 -0.35882 -0.150284 -0.02218  0.196198  0.030318
## 14  1.242212 -0.56161 -0.617922  0.08199  0.106896 -0.193683
## 15 -0.368970 -0.58101 -0.007750 -0.05095 -0.062378  0.057150
## 16 -0.669563 -0.04282 -0.047768  0.21397  0.152919 -0.277781
## 17 -0.761438 -0.24526  0.029662  0.03186  0.056651 -0.066478
## 18 -0.425436 -0.04832 -0.024151  0.02310  0.017175  0.226641
## 19 -0.234255 -0.27163  0.068958 -0.19446 -0.054914  0.018765
## 20  0.028982 -0.19608 -0.178394  0.05594 -0.199762  0.196180
## 21 -0.338971 -0.32675  0.150791 -0.08567 -0.105597 -0.022958
## 22 -0.128236 -0.32483  0.196796 -0.01866 -0.024355  0.385547
## 23 -0.363328 -0.18509  0.059418 -0.09107  0.017646 -0.048482
## 24 -0.596633 -0.45527  0.025255 -0.01668  0.055688 -0.127478
## 25 -0.030951 -0.36593  0.256660 -0.43481 -0.257818 -0.221803
## 26  0.192175 -0.44357  0.021972 -0.10034  0.052713  0.006413
## 27  0.041707 -0.25523 -0.121742  0.14506 -0.148207  0.038812
## 28  0.076520 -0.45121 -0.037076 -0.03006  0.096141 -0.035485
## 29  0.088275 -0.39706  0.026841  0.03825  0.115360  0.051904
## 30 -0.162040 -0.39082 -0.260650  0.12857 -0.041694  0.066871
## 31  0.400213  0.29276  0.183117  0.16617 -0.022077  0.210915
## 32  0.194940  0.27892  0.011061  0.10721  0.008114 -0.041581
## 33  0.096061 -0.04411  0.261267  0.44553 -0.062379 -0.335050
## 34 -0.082955  0.21771  0.144274 -0.37113  0.087125 -0.031603
## 35 -0.004696 -0.19546  0.173180 -0.07966  0.013836 -0.003374
## 36  0.217794  0.35991 -0.002031 -0.01349 -0.044245 -0.008067
## 37  0.530344  0.16473 -0.072752  0.08482 -0.179419  0.092896
## 38  0.386837  0.15010  0.163132 -0.04098  0.156511 -0.096270
## 39  0.454369  0.32809  0.158876  0.05545  0.229962  0.159861
## 40  0.370033  0.18849  0.150137 -0.01702  0.220776 -0.060391
## 41  0.733763  0.09891  0.543673 -0.25807 -0.215391 -0.102342
## 42  0.751397  0.28756 -0.018136  0.04335 -0.448897 -0.150765
## 43  0.260255  0.27344  0.283415 -0.15403  0.134286  0.053973
## 44  0.626689  0.03673  0.109380  0.08025  0.171751 -0.177150
## 45  0.422041  0.66969 -0.305212 -0.28421  0.194072 -0.040262
## 46 -0.308450 -0.38097 -0.156424 -0.03333  0.027181 -0.080304
## 47 -0.090387 -0.08771 -0.051645 -0.04502  0.049220 -0.105366
## 48  0.243109  0.17544  0.055587  0.16297  0.129623 -0.029377
## 49 -0.098095 -0.03125 -0.176481 -0.08163 -0.086694  0.113537
## 50 -0.123741 -0.28603 -0.028502  0.04567 -0.089356  0.096000
## 51 -0.220621 -0.11869  0.207751  0.06654 -0.025200  0.046210
## 52  0.447317 -0.04547  0.241645 -0.09827 -0.339366  0.198683
## 53  0.154271 -0.24656  0.021007  0.13312  0.174027  0.094545
## 54  0.139250 -0.34166  0.111607  0.17523  0.117702  0.191989
summary(pca2,scaling=2)
## 
## Call:
## rda(X = wc, scale = TRUE) 
## 
## Partitioning of correlations:
##               Inertia Proportion
## Total              18          1
## Unconstrained      18          1
## 
## Eigenvalues, and their contribution to the correlations 
## 
## Importance of components:
##                          PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Eigenvalue            8.0611 3.4761 1.50819 1.22521 0.71533 0.65066 0.61984
## Proportion Explained  0.4478 0.1931 0.08379 0.06807 0.03974 0.03615 0.03444
## Cumulative Proportion 0.4478 0.6410 0.72474 0.79281 0.83255 0.86870 0.90314
##                           PC8    PC9    PC10     PC11     PC12     PC13
## Eigenvalue            0.43868 0.4085 0.28913 0.166975 0.118712 0.107401
## Proportion Explained  0.02437 0.0227 0.01606 0.009276 0.006595 0.005967
## Cumulative Proportion 0.92751 0.9502 0.96627 0.975542 0.982137 0.988104
##                           PC14     PC15    PC16      PC17      PC18
## Eigenvalue            0.087748 0.063765 0.04590 0.0107740 0.0059461
## Proportion Explained  0.004875 0.003543 0.00255 0.0005986 0.0003303
## Cumulative Proportion 0.992979 0.996521 0.99907 0.9996697 1.0000000
## 
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores:  5.531195 
## 
## 
## Species scores
## 
##           PC1      PC2      PC3      PC4      PC5      PC6
## cond -0.99702  0.34972 -0.45496 -0.04411  0.13442  0.08488
## pH    0.17684  0.29544  0.57043 -0.85920 -0.54326 -0.30724
## turb -0.52396 -0.94668 -0.34857 -0.47256 -0.03982  0.25901
## Alk  -0.63291  0.85155 -0.31425  0.13417  0.12058 -0.12839
## TSS  -0.53705 -0.77957 -0.43743 -0.64323 -0.04180  0.28207
## PO4  -0.05355 -0.59396  0.73584 -0.33539  0.79636 -0.11188
## NH4  -0.77319  0.03783 -0.68069 -0.06619  0.19035 -0.53093
## NO2  -0.78431 -0.58641 -0.15264 -0.08530 -0.10877 -0.62400
## NO3  -0.67655 -0.89920  0.22318  0.46162 -0.18975 -0.01087
## Br   -1.20613 -0.13602  0.08215  0.06930  0.01147  0.28837
## Cl   -1.08187 -0.16509  0.49846  0.21967  0.03718 -0.12252
## Fl   -1.11621 -0.11372  0.18254 -0.05032  0.14197 -0.03546
## SO4  -1.05771  0.24288  0.40207  0.23188 -0.21474  0.01286
## Na   -1.14870  0.17217  0.13648  0.05892 -0.21020  0.18551
## K    -1.03824  0.48474  0.25134 -0.15451  0.03470  0.09851
## Ca   -0.95603  0.73948 -0.06107 -0.22861  0.11002  0.06037
## Mg   -0.99397  0.64159  0.05003 -0.24907  0.04694  0.06880
## TDN  -0.86580 -0.78875 -0.05764  0.31464 -0.19754 -0.07204
## 
## 
## Site scores (weighted sums of species scores)
## 
##          PC1      PC2       PC3      PC4      PC5      PC6
## 1  -1.283073  1.46362  0.329969  1.17010  0.42054  0.64811
## 2  -1.034856  1.29806  0.047297 -1.40897  0.23982  1.05835
## 3  -0.882710  1.69037 -0.205471  0.53435 -0.97579 -0.24031
## 4  -0.994400  0.94315  0.505607  0.14215 -0.25215 -0.60337
## 5  -2.431354  0.65908 -1.898250 -0.84073 -0.96461 -1.22401
## 7   0.625497  0.59533  0.335832  0.26999  0.46695 -1.53700
## 8   1.479591  0.71474 -2.670917 -1.30598 -0.33714  0.79142
## 9  -0.445864  0.46757 -0.662112  2.69914 -0.97712  0.59770
## 10 -0.761641  0.42997  0.078779 -1.25501  1.05851  0.23628
## 11  0.236561  0.72111 -0.598763  0.65145  1.19091  0.35206
## 12  1.141965  0.47600 -0.093490  0.25066 -0.74183 -0.50609
## 13 -0.266793 -0.81652 -0.519182 -0.08502  0.98419  0.15946
## 14  1.856240 -1.27798 -2.134725  0.31427  0.53622 -1.01871
## 15 -0.551353 -1.32214 -0.026774 -0.19529 -0.31291  0.30059
## 16 -1.000529 -0.09745 -0.165024  0.82015  0.76709 -1.46104
## 17 -1.137819 -0.55812  0.102472  0.12211  0.28418 -0.34966
## 18 -0.635730 -0.10995 -0.083433  0.08852  0.08615  1.19206
## 19 -0.350048 -0.61812  0.238228 -0.74534 -0.27547  0.09870
## 20  0.043308 -0.44619 -0.616293  0.21443 -1.00207  1.03184
## 21 -0.506526 -0.74355  0.520934 -0.32837 -0.52971 -0.12075
## 22 -0.191623 -0.73919  0.679868 -0.07152 -0.12217  2.02785
## 23 -0.542921 -0.42120  0.205271 -0.34908  0.08852 -0.25500
## 24 -0.891549 -1.03600  0.087247 -0.06393  0.27935 -0.67049
## 25 -0.046249 -0.83270  0.886679 -1.66658 -1.29329 -1.16661
## 26  0.287167 -1.00938  0.075905 -0.38458  0.26442  0.03373
## 27  0.062323 -0.58078 -0.420579  0.55599 -0.74345  0.20414
## 28  0.114344 -1.02675 -0.128087 -0.11522  0.48227 -0.18664
## 29  0.131909 -0.90355  0.092727  0.14660  0.57868  0.27300
## 30 -0.242136 -0.88933 -0.900465  0.49281 -0.20915  0.35172
## 31  0.598040  0.66619  0.632611  0.63693 -0.11075  1.10934
## 32  0.291300  0.63471  0.038214  0.41094  0.04070 -0.21870
## 33  0.143544 -0.10039  0.902594  1.70768 -0.31291 -1.76226
## 34 -0.123959  0.49543  0.498420 -1.42250  0.43705 -0.16622
## 35 -0.007017 -0.44478  0.598281 -0.30532  0.06941 -0.01775
## 36  0.325450  0.81900 -0.007016 -0.05172 -0.22195 -0.04243
## 37  0.792494  0.37487 -0.251334  0.32512 -0.90002  0.48860
## 38  0.578051  0.34156  0.563571 -0.15707  0.78511 -0.50635
## 39  0.678964  0.74658  0.548865  0.21253  1.15356  0.84082
## 40  0.552941  0.42893  0.518675 -0.06526  1.10748 -0.31764
## 41  1.096463  0.22508  1.878217 -0.98915 -1.08047 -0.53829
## 42  1.122814  0.65436 -0.062654  0.16617 -2.25180 -0.79297
## 43  0.388900  0.62223  0.979109 -0.59038  0.67362  0.28388
## 44  0.936463  0.08358  0.377874  0.30759  0.86156 -0.93175
## 45  0.630656  1.52392 -1.054410 -1.08937  0.97352 -0.21177
## 46 -0.460918 -0.86694 -0.540394 -0.12777  0.13635 -0.42237
## 47 -0.135065 -0.19959 -0.178417 -0.17257  0.24690 -0.55419
## 48  0.363278  0.39923  0.192034  0.62466  0.65023 -0.15452
## 49 -0.146583 -0.07110 -0.609688 -0.31289 -0.43489  0.59717
## 50 -0.184906 -0.65088 -0.098466  0.17505 -0.44824  0.50493
## 51 -0.329674 -0.27009  0.717715  0.25504 -0.12641  0.24305
## 52  0.668426 -0.10346  0.834806 -0.37667 -1.70236  1.04501
## 53  0.230527 -0.56106  0.072573  0.51022  0.87297  0.49728
## 54  0.208081 -0.77747  0.385568  0.67163  0.59043  1.00980

layout(matrix(1:2, nrow=1))
biplot(pca2,scaling=1)
biplot(pca2,scaling=2)

# note different scaling factors, but solution remains same


##############################
# some follow-up suggestions #
# test PCA-axes for effect of landuse or stream size (as log(Q)) using ANOVA or ANCOVA
# correlate PCA-axes with other potential "controlling" variables (e.g. canopy cover) to give "meta-dimensions" more meaning
# useful function envfit() to relate additional variables to the ordination space

zwc<-scale(wc) # must at least be centered even if dimensionally homogeneous!
xmat<-data.frame(logQ=log(mara.raw$Q),logTDN=log(mara.raw$TDN),canopy=mara.raw$canopy)[-6,]

rda<-rda(zwc~logQ+logTDN+canopy,data=xmat) # take care: confusing X and Y argument names

# actual RDA output check
summary(rda)
## 
## Call:
## rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat) 
## 
## Partitioning of variance:
##               Inertia Proportion
## Total          18.000       1.00
## Constrained     6.481       0.36
## Unconstrained  11.519       0.64
## 
## Eigenvalues, and their contribution to the variance 
## 
## Importance of components:
##                         RDA1    RDA2    RDA3    PC1     PC2     PC3     PC4
## Eigenvalue            5.5244 0.76796 0.18831 4.9837 1.63026 1.19739 0.77385
## Proportion Explained  0.3069 0.04266 0.01046 0.2769 0.09057 0.06652 0.04299
## Cumulative Proportion 0.3069 0.34957 0.36004 0.6369 0.72748 0.79400 0.83699
##                           PC5     PC6     PC7     PC8     PC9     PC10     PC11
## Eigenvalue            0.64230 0.60556 0.46542 0.38373 0.25602 0.153021 0.117674
## Proportion Explained  0.03568 0.03364 0.02586 0.02132 0.01422 0.008501 0.006537
## Cumulative Proportion 0.87268 0.90632 0.93217 0.95349 0.96772 0.976216 0.982754
##                           PC12     PC13     PC14     PC15      PC16      PC17
## Eigenvalue            0.101408 0.079286 0.061438 0.046321 0.0156525 0.0063286
## Proportion Explained  0.005634 0.004405 0.003413 0.002573 0.0008696 0.0003516
## Cumulative Proportion 0.988387 0.992792 0.996205 0.998779 0.9996484 1.0000000
## 
## Accumulated constrained eigenvalues
## Importance of components:
##                         RDA1   RDA2    RDA3
## Eigenvalue            5.5244 0.7680 0.18831
## Proportion Explained  0.8524 0.1185 0.02906
## Cumulative Proportion 0.8524 0.9709 1.00000
## 
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores:  5.531195 
## 
## 
## Species scores
## 
##         RDA1      RDA2      RDA3        PC1        PC2        PC3
## cond -0.6632  0.526208 -0.053197 -7.126e-01 -2.642e-01 -2.366e-01
## pH    0.4359 -0.080604 -0.110311 -2.354e-01  6.035e-01 -2.708e-01
## turb -0.8597  0.004151  0.171732  3.490e-01  4.901e-01 -5.836e-01
## Alk  -0.1085  0.409957 -0.026803 -8.856e-01 -5.159e-01 -4.483e-02
## TSS  -0.7648  0.195626  0.067225  2.585e-01  4.908e-01 -7.306e-01
## PO4  -0.1714 -0.057514 -0.327410  2.628e-01  9.564e-01  3.017e-01
## NH4  -0.6307  0.217307 -0.051937 -4.163e-01 -3.196e-01 -5.981e-01
## NO2  -0.8571 -0.178053  0.281840 -1.528e-01  2.746e-01 -3.129e-01
## NO3  -1.1170 -0.494302 -0.147602  1.649e-01  1.328e-01  3.170e-01
## Br   -0.9585  0.116087 -0.072049 -6.810e-01  2.180e-01  6.036e-02
## Cl   -0.8351 -0.016235  0.006982 -6.587e-01  4.095e-01  4.346e-01
## Fl   -0.7748  0.116617 -0.017050 -7.011e-01  4.096e-01  3.659e-02
## SO4  -0.5605  0.008991  0.140803 -9.589e-01  2.172e-01  3.459e-01
## Na   -0.7506  0.071258  0.039663 -8.902e-01  1.087e-01  6.183e-02
## K    -0.4348  0.085161 -0.092075 -1.085e+00  1.745e-01 -4.157e-02
## Ca   -0.3188  0.430074 -0.088844 -1.078e+00 -7.507e-02 -1.775e-01
## Mg   -0.3587  0.282375 -0.044108 -1.095e+00  4.243e-02 -1.784e-01
## TDN  -1.2252 -0.429552 -0.118224  2.170e-16 -1.802e-16  8.924e-17
## 
## 
## Site scores (weighted sums of species scores)
## 
##         RDA1      RDA2     RDA3      PC1      PC2       PC3
## 1  -0.899757  3.084889 -0.55294 -1.73973 -0.49579  1.405403
## 2  -0.601643  4.018791  0.23349 -1.62902  1.01934 -0.493815
## 3  -0.425217  2.834368  0.10401 -1.40811 -1.10996  0.502236
## 4  -0.756630  1.881205 -1.51654 -1.37687 -0.19517  0.362439
## 5  -2.455504  4.254210  2.42612 -2.05980 -0.84349 -2.485533
## 7   0.861601 -0.365693 -1.72079 -0.14991 -0.56146  0.155296
## 8   1.764345  1.159717  2.64686  0.62914 -1.54289 -2.968104
## 9  -0.482278  0.089585  0.15601 -0.35169 -2.59873  1.153105
## 10 -0.595642  2.398130 -0.19763 -0.94864  1.32755 -0.723328
## 11  0.451013  1.306409 -1.18506 -0.42811 -0.88023 -0.088124
## 12  1.394450 -0.925289  1.78413 -0.03078 -0.38908 -0.245790
## 13 -0.567171 -0.078456  0.53359  0.88444  0.28402  0.167429
## 14  1.469933 -2.003658  3.48847  3.00526 -1.06668 -0.266833
## 15 -1.033131 -1.201150  0.27119  0.59853  0.67023 -0.195910
## 16 -1.140389  0.708121  0.03320 -0.37468 -0.34163  0.496573
## 17 -1.426847  0.273804 -0.49117 -0.56728  0.24291 -0.207985
## 18 -0.726797  0.919409 -0.63159 -0.09927  0.15910  0.396983
## 19 -0.562127 -0.441430 -0.39299  0.45991  0.63986 -0.006655
## 20 -0.146589 -0.580216  1.46528  0.62302 -0.45546 -0.030056
## 21 -0.792760 -0.936524 -1.52046  0.06731  0.43021 -0.279145
## 22 -0.450183 -0.920766 -1.69925  0.35532  0.44301  0.394593
## 23 -0.722434 -0.099725 -1.05913  0.11364  0.32640  0.053559
## 24 -1.294677 -0.621767  0.54340 -0.14052  0.78301 -0.387924
## 25 -0.256636 -1.469854 -1.00794  0.30718  1.42878 -0.901286
## 26 -0.006011 -1.524849  0.44526  0.71389  0.74750 -0.370880
## 27 -0.167762 -1.210237  1.92863  0.73162 -0.47637  0.287856
## 28 -0.214533 -1.385767  0.40575  0.68065  0.48451 -0.293713
## 29 -0.153241 -1.359169  0.41669  0.54181  0.45796  0.113385
## 30 -0.608704 -0.767504  2.40998  0.58176 -0.34287 -0.271359
## 31  0.877051  0.127594 -0.99204 -0.11448 -0.08793  1.149609
## 32  0.524430  0.342174  0.11586 -0.19013 -0.41330  0.388517
## 33  0.100447 -1.912077 -0.27986 -0.07708 -0.63553  1.401936
## 34  0.113385  1.339155 -0.76471 -0.15951  1.13570 -0.158967
## 35 -0.144093 -0.913297 -2.68176  0.14873  0.34036 -0.116528
## 36  0.635959  0.706963 -0.20569 -0.45058 -0.27741 -0.210309
## 37  0.964039 -0.502987  1.56066  0.58492 -0.58515  0.246218
## 38  0.780714 -0.223068 -1.48202 -0.02480  0.36112  0.196388
## 39  1.002097  0.320595 -2.44794 -0.11533 -0.06812  0.544981
## 40  0.777442 -0.009334 -1.95549 -0.09691  0.23264  0.233320
## 41  1.374032 -1.702035  0.01784 -0.03704  1.50770  0.434674
## 42  1.421599 -1.106980  2.45558 -0.07522 -0.69935 -0.519837
## 43  0.700872  0.512436 -1.78976 -0.23677  0.87534  0.442824
## 44  1.074898 -1.074960  0.76215  0.24269  0.26216  0.475054
## 45  1.160824  2.438268 -2.38973  0.14196 -1.14818 -0.702500
## 46 -0.808194 -0.336965  0.64062 -0.06377  0.17315 -1.013755
## 47 -0.197930  0.127281  0.76092 -0.11609  0.31741 -0.241202
## 48  0.528880 -0.060544  0.32122  0.08513 -0.24501  0.789760
## 49 -0.190294  0.489138  1.74631 -0.11103  0.04844 -0.769059
## 50 -0.443365 -0.924883 -0.08718  0.49389 -0.16078  0.116308
## 51 -0.445001 -0.638721 -2.14463  0.13713  0.06390  0.808019
## 52  0.728592 -1.401673  2.01083  0.17276  0.63744  0.369228
## 53  0.045616 -1.090325 -1.06195  0.41535 -0.04511  0.260961
## 54 -0.036682 -1.542340  0.57423  0.45711  0.26592  0.601943
## 
## 
## Site constraints (linear combinations of constraining variables)
## 
##         RDA1     RDA2     RDA3      PC1      PC2       PC3
## 1  -0.192647  0.93038  0.92799 -1.73973 -0.49579  1.405403
## 2   0.427478  1.34335  1.22481 -1.62902  1.01934 -0.493815
## 3  -0.146209  0.99448  0.45228 -1.40811 -1.10996  0.502236
## 4  -0.310322 -0.13504  0.14698 -1.37687 -0.19517  0.362439
## 5  -1.484641 -0.21700  0.38344 -2.05980 -0.84349 -2.485533
## 7   0.674710 -0.78138 -0.51445 -0.14991 -0.56146  0.155296
## 8   1.388710 -0.10748 -0.22376  0.62914 -1.54289 -2.968104
## 9  -0.858711 -0.86050 -1.00156 -0.35169 -2.59873  1.153105
## 10  0.407941  1.11874 -0.68765 -0.94864  1.32755 -0.723328
## 11  0.703912 -0.23995 -0.63485 -0.42811 -0.88023 -0.088124
## 12  1.393070 -1.09377  1.50118 -0.03078 -0.38908 -0.245790
## 13 -0.864232  1.33162 -0.10131  0.88444  0.28402  0.167429
## 14  0.070930  1.59769  0.43814  3.00526 -1.06668 -0.266833
## 15 -1.113644 -0.02139  0.08487  0.59853  0.67023 -0.195910
## 16 -0.972647  0.47906 -0.44170 -0.37468 -0.34163  0.496573
## 17 -1.040565 -0.76624 -0.16526 -0.56728  0.24291 -0.207985
## 18 -0.629396  0.77943  0.19352 -0.09927  0.15910  0.396983
## 19 -0.880788  0.52142  0.58591  0.45991  0.63986 -0.006655
## 20 -0.557724  0.29157  0.85510  0.62302 -0.45546 -0.030056
## 21 -0.816524 -0.74571 -1.49309  0.06731  0.43021 -0.279145
## 22 -0.679726 -0.61138 -0.94934  0.35532  0.44301  0.394593
## 23 -0.876393  0.22614  0.17538  0.11364  0.32640  0.053559
## 24 -0.835658 -0.46318 -0.11007 -0.14052  0.78301 -0.387924
## 25 -0.306131 -0.56840 -0.71753  0.30718  1.42878 -0.901286
## 26 -0.050644 -0.33280 -0.12269  0.71389  0.74750 -0.370880
## 27 -0.663953  0.25632  0.79512  0.73162 -0.47637  0.287856
## 28 -0.304155 -0.31610  0.19760  0.68065  0.48451 -0.293713
## 29 -0.229915 -0.50523  0.78395  0.54181  0.45796  0.113385
## 30 -0.750822  0.01951  0.79961  0.58176 -0.34287 -0.271359
## 31  0.919784  0.24149 -0.12906 -0.11448 -0.08793  1.149609
## 32  0.500061  0.28682  0.07581 -0.19013 -0.41330  0.388517
## 33 -0.169995 -1.32278  0.05458 -0.07708 -0.63553  1.401936
## 34  0.327650  1.71656 -0.89375 -0.15951  1.13570 -0.158967
## 35 -0.278900 -0.83339 -1.12100  0.14873  0.34036 -0.116528
## 36  0.827677  0.04073 -0.62665 -0.45058 -0.27741 -0.210309
## 37  0.573226  0.88396 -1.00915  0.58492 -0.58515  0.246218
## 38  0.850153 -0.05214 -0.58813 -0.02480  0.36112  0.196388
## 39  1.044336  0.18841 -1.50261 -0.11533 -0.06812  0.544981
## 40  0.840242 -0.12799 -0.57504 -0.09691  0.23264  0.233320
## 41  1.451280 -0.87165  0.50304 -0.03704  1.50770  0.434674
## 42  1.324854 -1.06452 -0.01889 -0.07522 -0.69935 -0.519837
## 43  0.954237  0.72568 -1.21010 -0.23677  0.87534  0.442824
## 44  1.025094 -0.47834  1.59291  0.24269  0.26216  0.475054
## 45  0.465138  1.60579  0.20294  0.14196 -1.14818 -0.702500
## 46 -0.461071 -1.14745 -0.17978 -0.06377  0.17315 -1.013755
## 47  0.002323 -0.19579  1.34466 -0.11609  0.31741 -0.241202
## 48  0.426895  0.53629 -0.03598  0.08513 -0.24501  0.789760
## 49  0.119534  0.16802 -0.14099 -0.11103  0.04844 -0.769059
## 50 -0.832201 -0.18606  0.31023  0.49389 -0.16078  0.116308
## 51 -0.803215 -0.10485 -0.18123  0.13713  0.06390  0.808019
## 52  0.580101 -0.79935  1.87566  0.17276  0.63744  0.369228
## 53 -0.091596 -0.62895 -0.07307  0.41535 -0.04511  0.260961
## 54 -0.096912 -0.70466 -0.05704  0.45711  0.26592  0.601943
## 
## 
## Biplot scores for constraining variables
## 
##           RDA1    RDA2     RDA3 PC1 PC2 PC3
## logQ    0.5174 -0.6315  0.57757   0   0   0
## logTDN -0.9398 -0.3295 -0.09068   0   0   0
## canopy  0.5328 -0.2110 -0.81953   0   0   0
str(rda)
## List of 12
##  $ colsum        : Named num [1:18] 1 1 1 1 1 1 1 1 1 1 ...
##   ..- attr(*, "names")= chr [1:18] "cond" "pH" "turb" "Alk" ...
##  $ tot.chi       : num 18
##  $ Ybar          : num [1:53, 1:18] 0.248 0.215 0.216 0.178 0.42 ...
##   ..- attr(*, "scaled:center")= Named num [1:18] 8.64e-16 -1.43e-15 -1.05e-18 -7.64e-16 -1.31e-16 ...
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##   .. ..- attr(*, "names")= chr [1:18] "cond" "pH" "turb" "Alk" ...
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##   .. ..$ : chr [1:18] "cond" "pH" "turb" "Alk" ...
##   ..- attr(*, "METHOD")= chr "PCA"
##  $ method        : chr "rda"
##  $ call          : language rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##  $ pCCA          : NULL
##  $ CCA           :List of 12
##   ..$ eig      : Named num [1:3] 5.524 0.768 0.188
##   .. ..- attr(*, "names")= chr [1:3] "RDA1" "RDA2" "RDA3"
##   ..$ poseig   : NULL
##   ..$ u        : num [1:53, 1:3] -0.0348 0.0773 -0.0264 -0.0561 -0.2684 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:53] "1" "2" "3" "4" ...
##   .. .. ..$ : chr [1:3] "RDA1" "RDA2" "RDA3"
##   ..$ v        : num [1:18, 1:3] -0.2164 0.1423 -0.2806 -0.0354 -0.2496 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:18] "cond" "pH" "turb" "Alk" ...
##   .. .. ..$ : chr [1:3] "RDA1" "RDA2" "RDA3"
##   ..$ wa       : num [1:53, 1:3] -0.1627 -0.1088 -0.0769 -0.1368 -0.4439 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:53] "1" "2" "3" "4" ...
##   .. .. ..$ : chr [1:3] "RDA1" "RDA2" "RDA3"
##   ..$ alias    : NULL
##   ..$ biplot   : num [1:3, 1:3] 0.517 -0.94 0.533 -0.631 -0.329 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:3] "logQ" "logTDN" "canopy"
##   .. .. ..$ : chr [1:3] "RDA1" "RDA2" "RDA3"
##   ..$ rank     : int 3
##   ..$ qrank    : int 3
##   ..$ tot.chi  : num 6.48
##   ..$ QR       :List of 4
##   .. ..$ qr   : num [1:53, 1:3] 14.67002 -0.01451 0.07998 -0.00174 0.07405 ...
##   .. .. ..- attr(*, "assign")= int [1:3] 1 2 3
##   .. .. ..- attr(*, "dimnames")=List of 2
##   .. .. .. ..$ : chr [1:53] "1" "2" "3" "4" ...
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##   .. .. ..- attr(*, "scaled:center")= Named num [1:3] 1.674 -9.205 0.464
##   .. .. .. ..- attr(*, "names")= chr [1:3] "logQ" "logTDN" "canopy"
##   .. ..$ rank : int 3
##   .. ..$ qraux: num [1:3] 1.03 1.18 1.13
##   .. ..$ pivot: int [1:3] 1 2 3
##   .. ..- attr(*, "class")= chr "qr"
##   ..$ envcentre: Named num [1:3] 1.674 -9.205 0.464
##   .. ..- attr(*, "names")= chr [1:3] "logQ" "logTDN" "canopy"
##  $ CA            :List of 6
##   ..$ eig    : Named num [1:17] 4.984 1.63 1.197 0.774 0.642 ...
##   .. ..- attr(*, "names")= chr [1:17] "PC1" "PC2" "PC3" "PC4" ...
##   ..$ poseig : NULL
##   ..$ u      : num [1:53, 1:17] -0.315 -0.295 -0.255 -0.249 -0.372 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:53] "1" "2" "3" "4" ...
##   .. .. ..$ : chr [1:17] "PC1" "PC2" "PC3" "PC4" ...
##   ..$ v      : num [1:18, 1:17] -0.2449 -0.0809 0.1199 -0.3043 0.0888 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:18] "cond" "pH" "turb" "Alk" ...
##   .. .. ..$ : chr [1:17] "PC1" "PC2" "PC3" "PC4" ...
##   ..$ rank   : int 17
##   ..$ tot.chi: num 11.5
##  $ inertia       : chr "variance"
##  $ regularization: chr "this is a vegan::rda result object"
##  $ terms         :Classes 'terms', 'formula'  language zwc ~ logQ + logTDN + canopy
##   .. ..- attr(*, "variables")= language list(zwc, logQ, logTDN, canopy)
##   .. ..- attr(*, "factors")= int [1:4, 1:3] 0 1 0 0 0 0 1 0 0 0 ...
##   .. .. ..- attr(*, "dimnames")=List of 2
##   .. .. .. ..$ : chr [1:4] "zwc" "logQ" "logTDN" "canopy"
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##   .. ..- attr(*, "term.labels")= chr [1:3] "logQ" "logTDN" "canopy"
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##   .. ..- attr(*, "response")= int 1
##   .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> 
##  $ terminfo      :List of 4
##   ..$ terms  :Classes 'terms', 'formula'  language ~logQ + logTDN + canopy
##   .. .. ..- attr(*, "variables")= language list(logQ, logTDN, canopy)
##   .. .. ..- attr(*, "factors")= int [1:3, 1:3] 1 0 0 0 1 0 0 0 1
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##   .. .. .. .. ..$ : chr [1:3] "logQ" "logTDN" "canopy"
##   .. .. ..- attr(*, "term.labels")= chr [1:3] "logQ" "logTDN" "canopy"
##   .. .. ..- attr(*, "order")= int [1:3] 1 1 1
##   .. .. ..- attr(*, "intercept")= int 1
##   .. .. ..- attr(*, "response")= num 0
##   .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> 
##   ..$ assign : int [1:3] 1 2 3
##   ..$ xlev   : Named list()
##   ..$ ordered: Named logi [1:3] FALSE FALSE FALSE
##   .. ..- attr(*, "names")= chr [1:3] "logQ" "logTDN" "canopy"
##  - attr(*, "class")= chr [1:2] "rda" "cca"
?cca.object
## starte den http Server für die Hilfe fertig
RsquareAdj(rda) # redundancy statistic (fractional amount of variation of the response data matrix explained by constraints)
## $r.squared
## [1] 0.3600362
## 
## $adj.r.squared
## [1] 0.3208547

# hypothesis tests #
# testing the first axis (global test)
anova(rda)
## Permutation test for rda under reduced model
## Permutation: free
## Number of permutations: 999
## 
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##          Df Variance      F Pr(>F)    
## Model     3   6.4807 9.1889  0.001 ***
## Residual 49  11.5193                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(rda,first=TRUE)
## Permutation test for rda under reduced model
## Permutation: free
## Number of permutations: 999
## 
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##          Df Variance      F Pr(>F)    
## RDA1      1   5.5244 23.499  0.001 ***
## Residual 49  11.5193                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# testing all axes sequentially (preceding axes are taken as constraints)
anova(rda,by="axis",model="direct",perm.max=9999,step=1000)
## Permutation test for rda under direct model
## Forward tests for axes
## Permutation: free
## Number of permutations: 999
## 
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##          Df Variance       F Pr(>F)    
## RDA1      1   5.5244 23.4991  0.001 ***
## RDA2      1   0.7680  3.2667  0.050 *  
## RDA3      1   0.1883  0.8010  0.518    
## Residual 49  11.5193                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# testing the individual terms=constraints
anova(rda,by="terms",model="direct",perm.max=9999,step=1000)  # tests terms sequentially, order matters!
## Permutation test for rda under direct model
## Terms added sequentially (first to last)
## Permutation: free
## Number of permutations: 999
## 
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##          Df Variance       F Pr(>F)    
## logQ      1   1.8477  7.8596  0.001 ***
## logTDN    1   3.9175 16.6639  0.001 ***
## canopy    1   0.7155  3.0434  0.021 *  
## Residual 49  11.5193                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(rda,by="margin",model="direct",perm.max=9999,step=1000) # tests each term in full model (like drop1() function)
## Permutation test for rda under direct model
## Marginal effects of terms
## Permutation: free
## Number of permutations: 999
## 
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##          Df Variance       F Pr(>F)    
## logQ      1   1.0182  4.3312  0.007 ** 
## logTDN    1   2.6904 11.4441  0.001 ***
## canopy    1   0.7155  3.0434  0.027 *  
## Residual 49  11.5193                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

###################
# making triplots #

# again various types of scaling for the plotting step:
# scaling 1 "distance triplot"
# only angles between constraints and responses reflect their correlations (not angles among responses)
# distances among sites reflect their Euclidean distances

# scaling 2 "correlation triplot"
# all angles between constraints and responses reflect correlations 
# distances among sites do not reflect their Euclidean distances

# in both scaling types sites can be projected on constraints and on responses
# factor constraints are shown as centroids instead of arrows, projecting works identical 

# scaling 3 is compromise

# build an RDA scaling type 1 triplot
plot(rda,scaling=1)

(sites<-scores(rda,choices=c(1,2),display="sites",scaling=1)) 
##            RDA1         RDA2
## 1  -0.498460582  0.637194072
## 2  -0.333307201  0.830094727
## 3  -0.235567821  0.585448173
## 4  -0.419169104  0.388569136
## 5  -1.360336512  0.878721129
## 7   0.477322538 -0.075535075
## 8   0.977437916  0.239543485
## 9  -0.267179489  0.018503982
## 10 -0.329982722  0.495341683
## 11  0.249858739  0.269843172
## 12  0.772517930 -0.191121436
## 13 -0.314209914 -0.016205304
## 14  0.814335423 -0.413862255
## 15 -0.572349386 -0.248101430
## 16 -0.631769326  0.146264714
## 17 -0.790465648  0.056555154
## 18 -0.402641922  0.189907030
## 19 -0.311415221 -0.091178758
## 20 -0.081209267 -0.119845614
## 21 -0.439185103 -0.193442177
## 22 -0.249398850 -0.190187243
## 23 -0.400224485 -0.020598570
## 24 -0.717244600 -0.128427989
## 25 -0.142174974 -0.303603256
## 26 -0.003329834 -0.314962678
## 27 -0.092939197 -0.249978470
## 28 -0.118850427 -0.286234700
## 29 -0.084894890 -0.280740961
## 30 -0.337218829 -0.158530543
## 31  0.485881970  0.026354971
## 32  0.290531318  0.070677146
## 33  0.055647198 -0.394945899
## 34  0.062814787  0.276606892
## 35 -0.079826826 -0.188644454
## 36  0.352318114  0.146025624
## 37  0.534072469 -0.103893687
## 38  0.432511681 -0.046075409
## 39  0.555156725  0.066219970
## 40  0.430698723 -0.001927959
## 41  0.761206836 -0.351560915
## 42  0.787558621 -0.228650302
## 43  0.388279575  0.105845304
## 44  0.595488024 -0.222036636
## 45  0.643090631  0.503632310
## 46 -0.447735520 -0.069601236
## 47 -0.109651939  0.026290352
## 48  0.292996889 -0.012505478
## 49 -0.105422063  0.101033121
## 50 -0.245621890 -0.191037745
## 51 -0.246528053 -0.131929977
## 52  0.403636317 -0.289520170
## 53  0.025270757 -0.225210312
## 54 -0.020321586 -0.318575509
## attr(,"const")
## [1] 5.531195

(lcs<-scores(rda,choices=c(1,2),display="lc",scaling=1)) # fitted/constrained site scores
##            RDA1         RDA2
## 1  -0.106725620  0.192172696
## 2   0.236820410  0.277474421
## 3  -0.080999194  0.205412919
## 4  -0.171916633 -0.027892958
## 5  -0.822483293 -0.044821584
## 7   0.373785769 -0.161396377
## 8   0.769338143 -0.022200417
## 9  -0.475721291 -0.177739370
## 10  0.225997452  0.231079381
## 11  0.389963824 -0.049561659
## 12  0.771753557 -0.225921755
## 13 -0.478779804  0.275051081
## 14  0.039294802  0.330009134
## 15 -0.616953195 -0.004418402
## 16 -0.538841264  0.098951894
## 17 -0.576467626 -0.158269346
## 18 -0.348682347  0.160994450
## 19 -0.487951895  0.107700881
## 20 -0.308976357  0.060225476
## 21 -0.452349895 -0.154028956
## 22 -0.376564556 -0.126282122
## 23 -0.485517314  0.046710238
## 24 -0.462950388 -0.095671797
## 25 -0.169594740 -0.117405573
## 26 -0.028056760 -0.068740794
## 27 -0.367826399  0.052943956
## 28 -0.168500544 -0.065292121
## 29 -0.127371851 -0.104357404
## 30 -0.415951385  0.004030800
## 31  0.509555593  0.049880617
## 32  0.277030998  0.059243495
## 33 -0.094176545 -0.273225329
## 34  0.181516631  0.354560391
## 35 -0.154509114 -0.172139937
## 36  0.458528832  0.008412040
## 37  0.317564066  0.182584970
## 38  0.470980327 -0.010770442
## 39  0.578556877  0.038917612
## 40  0.465489775 -0.026436288
## 41  0.804001463 -0.180041364
## 42  0.733962381 -0.219879621
## 43  0.528642419  0.149890940
## 44  0.567896692 -0.098802741
## 45  0.257683843  0.331681637
## 46 -0.255430728 -0.237010393
## 47  0.001287162 -0.040440796
## 48  0.236497729  0.110772850
## 49  0.066221216  0.034706118
## 50 -0.461034789 -0.038432073
## 51 -0.444977082 -0.021657745
## 52  0.321373087 -0.165108083
## 53 -0.050743664 -0.129911681
## 54 -0.053688775 -0.145550870
## attr(,"const")
## [1] 5.531195

(species<-scores(rda,choices=c(1,2),display="sp",scaling=1)*0.5)
##             RDA1        RDA2
## cond -0.59858199  1.27378225
## pH    0.39342255 -0.19511683
## turb -0.77595131  0.01004908
## Alk  -0.09788046  0.99237637
## TSS  -0.69028937  0.47354883
## PO4  -0.15467198 -0.13922271
## NH4  -0.56926510  0.52603200
## NO2  -0.77353865 -0.43101017
## NO3  -1.00809786 -1.19654781
## Br   -0.86504284  0.28100954
## Cl   -0.75374148 -0.03929891
## Fl   -0.69926370  0.28229280
## SO4  -0.50586602  0.02176366
## Na   -0.67742612  0.17249327
## K    -0.39244249  0.20614834
## Ca   -0.28777201  1.04107246
## Mg   -0.32375066  0.68353998
## TDN  -1.10581086 -1.03980988
## attr(,"const")
## [1] 5.531195

(constraints<-scores(rda,choices=c(1,2),display="bp",scaling=1)*2)
##              RDA1       RDA2
## logQ    0.5732272 -0.2608649
## logTDN -1.0412848 -0.1361117
## canopy  0.5903155 -0.0871586
## attr(,"const")
## [1] 5.531195

plot(sites,asp=1,pch=21,bg=landuse,ylim=c(-1.5,1.5))
Arrows(x0=0,y0=0,x1=constraints[,1],y1=constraints[,2],lwd=1.5,col="blue")
text(constraints[,1:2]*1.1,label=rownames(constraints),pos=4,cex=0.8,col="blue")

Arrows(x0=0,y0=0,x1=species[,1],y1=species[,2],lwd=1,arr.length=0)
text(species[,1:2]*1.1,label=rownames(species),pos=4,cex=0.6)

data(varespec) # a R dataset on vegetation
data(varechem) # soil chemistry
head(varespec)
##    Callvulg Empenigr Rhodtome Vaccmyrt Vaccviti Pinusylv Descflex Betupube
## 18     0.55    11.13     0.00     0.00    17.80     0.07     0.00        0
## 15     0.67     0.17     0.00     0.35    12.13     0.12     0.00        0
## 24     0.10     1.55     0.00     0.00    13.47     0.25     0.00        0
## 27     0.00    15.13     2.42     5.92    15.97     0.00     3.70        0
## 23     0.00    12.68     0.00     0.00    23.73     0.03     0.00        0
## 19     0.00     8.92     0.00     2.42    10.28     0.12     0.02        0
##    Vacculig Diphcomp Dicrsp Dicrfusc Dicrpoly Hylosple Pleuschr Polypili
## 18     1.60     2.07   0.00     1.62     0.00      0.0     4.67     0.02
## 15     0.00     0.00   0.33    10.92     0.02      0.0    37.75     0.02
## 24     0.00     0.00  23.43     0.00     1.68      0.0    32.92     0.00
## 27     1.12     0.00   0.00     3.63     0.00      6.7    58.07     0.00
## 23     0.00     0.00   0.00     3.42     0.02      0.0    19.42     0.02
## 19     0.00     0.00   0.00     0.32     0.02      0.0    21.03     0.02
##    Polyjuni Polycomm Pohlnuta Ptilcili Barbhatc Cladarbu Cladrang Cladstel
## 18     0.13     0.00     0.13     0.12     0.00    21.73    21.47     3.50
## 15     0.23     0.00     0.03     0.02     0.00    12.05     8.13     0.18
## 24     0.23     0.00     0.32     0.03     0.00     3.58     5.52     0.07
## 27     0.00     0.13     0.02     0.08     0.08     1.42     7.63     2.55
## 23     2.12     0.00     0.17     1.80     0.02     9.08     9.22     0.05
## 19     1.58     0.18     0.07     0.27     0.02     7.23     4.95    22.08
##    Cladunci Cladcocc Cladcorn Cladgrac Cladfimb Cladcris Cladchlo Cladbotr
## 18     0.30     0.18     0.23     0.25     0.25     0.23     0.00     0.00
## 15     2.65     0.13     0.18     0.23     0.25     1.23     0.00     0.00
## 24     8.93     0.00     0.20     0.48     0.00     0.07     0.10     0.02
## 27     0.15     0.00     0.38     0.12     0.10     0.03     0.00     0.02
## 23     0.73     0.08     1.42     0.50     0.17     1.78     0.05     0.05
## 19     0.25     0.10     0.25     0.18     0.10     0.12     0.05     0.02
##    Cladamau Cladsp Cetreric Cetrisla Flavniva Nepharct Stersp Peltapht Icmaeric
## 18     0.08   0.02     0.02     0.00     0.12     0.02   0.62     0.02        0
## 15     0.00   0.00     0.15     0.03     0.00     0.00   0.85     0.00        0
## 24     0.00   0.00     0.78     0.12     0.00     0.00   0.03     0.00        0
## 27     0.00   0.02     0.00     0.00     0.00     0.00   0.00     0.07        0
## 23     0.00   0.00     0.00     0.00     0.02     0.00   1.58     0.33        0
## 19     0.00   0.00     0.00     0.00     0.02     0.00   0.28     0.00        0
##    Cladcerv Claddefo Cladphyl
## 18        0     0.25        0
## 15        0     1.00        0
## 24        0     0.33        0
## 27        0     0.15        0
## 23        0     1.97        0
## 19        0     0.37        0
head(varechem)
##       N    P     K    Ca    Mg    S    Al   Fe    Mn   Zn  Mo Baresoil Humdepth
## 18 19.8 42.1 139.9 519.4  90.0 32.3  39.0 40.9  58.1  4.5 0.3     43.9      2.2
## 15 13.4 39.1 167.3 356.7  70.7 35.2  88.1 39.0  52.4  5.4 0.3     23.6      2.2
## 24 20.2 67.7 207.1 973.3 209.1 58.1 138.0 35.4  32.1 16.8 0.8     21.2      2.0
## 27 20.6 60.8 233.7 834.0 127.2 40.7  15.4  4.4 132.0 10.7 0.2     18.7      2.9
## 23 23.8 54.5 180.6 777.0 125.8 39.5  24.2  3.0  50.1  6.6 0.3     46.0      3.0
## 19 22.8 40.9 171.4 691.8 151.4 40.8 104.8 17.6  43.6  9.1 0.4     40.5      3.8
##     pH
## 18 2.7
## 15 2.8
## 24 3.0
## 27 2.8
## 23 2.7
## 19 2.7
apply(varespec,1,sum) # approximate 100 (total cover), "absolute" abundance data
##     18     15     24     27     23     19     22     16     28     13     14 
##  89.20  89.82  94.21 125.61  90.46  81.27 109.76  88.52 110.70 101.89  81.65 
##     20     25      7      5      6      3      4      2      9     12     10 
##  64.11  94.06 103.38  94.77 110.90 106.67  84.83 119.13 122.60 119.80 122.37 
##     11     21 
## 112.84  99.17

# correspondence analysis #
# run a CA just based on the species data (unconstrained!)
vare.ca<-cca(X=varespec) # function also used for CCA, but here only one matrix X is supplied

summary(vare.ca,scaling=1)
## 
## Call:
## cca(X = varespec) 
## 
## Partitioning of scaled Chi-square:
##               Inertia Proportion
## Total           2.083          1
## Unconstrained   2.083          1
## 
## Eigenvalues, and their contribution to the scaled Chi-square 
## 
## Importance of components:
##                          CA1    CA2    CA3     CA4     CA5     CA6     CA7
## Eigenvalue            0.5249 0.3568 0.2344 0.19546 0.17762 0.12156 0.11549
## Proportion Explained  0.2520 0.1713 0.1125 0.09383 0.08526 0.05835 0.05544
## Cumulative Proportion 0.2520 0.4233 0.5358 0.62962 0.71489 0.77324 0.82868
##                           CA8     CA9    CA10    CA11    CA12    CA13     CA14
## Eigenvalue            0.08894 0.07318 0.05752 0.04434 0.02546 0.01710 0.014896
## Proportion Explained  0.04269 0.03513 0.02761 0.02129 0.01222 0.00821 0.007151
## Cumulative Proportion 0.87137 0.90650 0.93411 0.95539 0.96762 0.97583 0.982978
##                           CA15     CA16     CA17     CA18     CA19      CA20
## Eigenvalue            0.010160 0.007830 0.006032 0.004008 0.002865 0.0019275
## Proportion Explained  0.004877 0.003759 0.002896 0.001924 0.001375 0.0009253
## Cumulative Proportion 0.987855 0.991614 0.994510 0.996434 0.997809 0.9987341
##                            CA21      CA22      CA23
## Eigenvalue            0.0018074 0.0005864 0.0002434
## Proportion Explained  0.0008676 0.0002815 0.0001168
## Cumulative Proportion 0.9996017 0.9998832 1.0000000
## 
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
## 
## 
## Species scores
## 
##                 CA1       CA2      CA3       CA4        CA5       CA6
## Callvulg  0.0303167 -1.597460  0.11455 -2.894569  0.1376073  2.291129
## Empenigr  0.0751030  0.379305  0.39303  0.023675  0.8568729 -0.400964
## Rhodtome  1.1052309  1.499299  3.04284  0.120106  3.2324306 -0.283510
## Vaccmyrt  1.4614812  1.622935  2.72375  0.231688  0.4604556  0.712538
## Vaccviti  0.1468014  0.313436  0.14696  0.243505  0.6868371 -0.147815
## Pinusylv -0.4820096  0.588517 -0.36020 -0.127094  0.4064754  0.386604
## Descflex  1.5348239  1.218806  1.87562 -0.001340 -1.3136979 -0.070731
## Betupube  0.6694503  1.951826  3.84017  1.389423  7.5959115 -0.244478
## Vacculig -0.0830789 -1.629259  1.05063  0.802648 -0.3058811 -1.625341
## Diphcomp -0.5446464 -1.037570  0.52282  0.940275  0.3682126 -1.082929
## Dicrsp    1.8120408  0.360290 -4.92082  3.088562  1.3867372  0.157815
## Dicrfusc  1.2704743 -0.562978 -0.39718 -2.929542  0.3848272 -2.408710
## Dicrpoly  0.7248118  1.409347  0.80341  1.915549  4.5674148  1.295447
## Hylosple  2.0062408  1.743883  2.27549  0.928884 -3.7648428  2.254851
## Pleuschr  1.3102086  0.583036 -0.01004  0.137298 -1.1216144  0.200422
## Polypili -0.3805097 -1.243904  0.54593  1.477188 -0.7276341 -0.387641
## Polyjuni  1.0133795  0.099043 -2.24697  1.510641  0.7729714 -3.062378
## Polycomm  0.8468241  1.321773  1.13585  1.140723  2.6836594 -0.605038
## Pohlnuta -0.0136453  0.589290 -0.35542  0.135481  0.9369707  0.397246
## Ptilcili  0.4223631  1.598584  3.43474  1.400065  6.3209491  0.198935
## Barbhatc  0.5018348  2.119334  4.57303  1.693188  8.1101807  0.645995
## Cladarbu -0.1531729 -1.483884  0.20024  0.193680  0.0734141  0.358926
## Cladrang -0.5502561 -1.084008  0.40552  0.724060 -0.3357992 -0.335924
## Cladstel -1.4373146  1.077753 -0.44397 -0.375926 -0.2421525  0.004212
## Cladunci  0.8151727 -1.006186 -1.82587 -1.389523  1.6046713  3.675908
## Cladcocc -0.2133215 -0.584429 -0.21434 -0.567886 -0.0003788 -0.145303
## Cladcorn  0.2631227 -0.177858 -0.44464  0.272422  0.3992282 -0.306738
## Cladgrac  0.1956947 -0.311167 -0.23894  0.379013  0.4933026  0.037581
## Cladfimb  0.0009213 -0.161418  0.18463 -0.435908  0.4831233 -0.143751
## Cladcris  0.3373031 -0.470369 -0.05093 -0.823855  0.7182250  0.636140
## Cladchlo -0.6200021  1.207278  0.21889  0.426447  1.9506082  0.120722
## Cladbotr  0.5647242  1.047333  2.65330  0.907734  4.4946805  1.201655
## Cladamau -0.6598144 -1.512880  0.83251  1.577699 -0.0407227 -1.419139
## Cladsp   -0.8209003  0.476164 -0.49752 -0.998241 -0.2393208  0.390785
## Cetreric  0.2458192 -0.689228 -1.68427 -0.131681  0.7439412  2.374535
## Cetrisla -0.3465221  1.362693  0.85897  0.396752  2.7526968  0.396591
## Flavniva -1.4391907 -0.833589 -0.12919  0.007071 -1.4841375  2.956977
## Nepharct  1.6813309  0.199484 -4.33509  2.229917  0.9561223 -5.472858
## Stersp   -0.5172793 -2.280900  0.99775  2.377013 -0.8892757 -1.441228
## Peltapht  0.4035858 -0.043265  0.04538  0.711040  0.1824679 -0.841227
## Icmaeric  0.0378754 -2.419595  0.72135  0.361302 -0.3736424 -2.092136
## Cladcerv -0.9232858 -0.005233 -1.22058  0.305290 -0.8142627  0.414135
## Claddefo  0.5190399 -0.496632 -0.15271 -0.695927  0.9042143  0.909191
## Cladphyl -1.2836161  1.155872 -0.79912 -0.741170 -0.1608002  0.490526
## 
## 
## Site scores (weighted averages of species scores)
## 
##          CA1      CA2       CA3      CA4        CA5      CA6
## 18 -0.108122 -0.53705  0.229574  0.24412  0.1405624 -0.14253
## 15  0.697118 -0.14441 -0.031788 -0.21743 -0.2738522 -0.08146
## 24  0.987603  0.15042 -1.348447  0.80472  0.3095168  0.46773
## 27  0.851765  0.49901  0.443559  0.12277 -0.4814871  0.07589
## 23  0.359881 -0.05608  0.145813  0.15087  0.2405263 -0.17770
## 19  0.003545  0.37017  0.027760  0.06168 -0.1158930 -0.03413
## 22  0.860732 -0.11504  0.110869 -1.02169  0.0772348 -0.60530
## 16  0.636936 -0.33250  0.001120 -0.79797  0.0130769 -0.54049
## 28  1.279352  0.81557  0.670053  0.23137 -0.8929976  0.41783
## 13 -0.195009 -0.80564  0.117686 -0.58286 -0.0007212  0.53071
## 14  0.528532 -0.70420 -0.517771 -0.86836  0.5713441  0.91671
## 20  0.382866 -0.18686 -0.004789  0.10156  0.0458125  0.21087
## 25  0.990715  0.11967 -1.110040  0.44929  0.1885902 -0.70694
## 7  -0.264704 -1.06013  0.334900  0.45973 -0.0326631 -0.19945
## 5  -0.428410 -1.20765  0.374344  0.74970 -0.2596294 -0.30467
## 6  -0.330534 -0.77498  0.130760  0.22391  0.0632686  0.09060
## 3  -0.899601  0.12075 -0.075742  0.03842 -0.1489585 -0.12031
## 4  -0.770294 -0.35351 -0.033779 -0.01795 -0.3007839  0.44303
## 2  -0.992193  0.50319 -0.157505 -0.07070 -0.1065172 -0.09928
## 9  -0.937173  0.78688 -0.258119 -0.19377 -0.0343535 -0.01259
## 12 -0.726413  0.49163 -0.157235 -0.08698 -0.0105774 -0.02801
## 10 -1.002083  0.71239 -0.236526 -0.18643 -0.0231666 -0.04928
## 11 -0.322647 -0.03871 -0.001297  0.09029 -0.1481448  0.06934
## 21  0.259527  0.80746  1.124258  0.36083  1.5437866  0.07051
# summary(vare.ca,scaling=2)
# again two different types of scaling are possible for biplots

# scaling 1 (distances among sites matter)
# distances among sites approximate their chi^2 distance
# close sites have similar species abundances
# a site, which is near a specific species, has a high contribution of that species 

# scaling 2 (relationships among species matter)
# distances among species approximate their chi^2 distance
# close species have similar abundances across sites
# a species, which is near a specific site, is more likely to be found at that site

plot(vare.ca,scaling=1)

plot(vare.ca,scaling=2)

plot(vare.ca,scaling=3) # a compromise

# for any scaling take care when interpreting species close to origin:
# these are "everywhere" or have optimum right at the origin (i.e., optimum with regard to both axis shown)

# for more controlled plotting
species.scores<-scores(vare.ca,display="species",scaling=2)
site.scores<-scores(vare.ca,display="sites",scaling=2)

plot(site.scores,col="black",pch=21,xlim=c(-2,2),ylim=c(-2,2))
text(species.scores,col="red",label=names(varespec),cex=0.7)

# post-hoc fitting of an environmental variable
names(varechem)
##  [1] "N"        "P"        "K"        "Ca"       "Mg"       "S"       
##  [7] "Al"       "Fe"       "Mn"       "Zn"       "Mo"       "Baresoil"
## [13] "Humdepth" "pH"
(ef<-envfit(vare.ca,varechem[,12:13],permutations=1999))
## 
## ***VECTORS
## 
##               CA1      CA2     r2 Pr(>r)   
## Baresoil  0.97947 -0.20161 0.2533  0.058 . 
## Humdepth  0.91602  0.40112 0.4524  0.003 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Permutation: free
## Number of permutations: 1999
plot(ef)

###########################
# canonical correspondence analysis #
vare.cca<-cca(Y=varespec,X=varechem) # note strange terminology of X and Y in vegan (don´t ask)
vare.cca<-cca(varespec~.,varechem) # hypothesis tests need formula interface (don´t ask)

summary(vare.cca,scaling=1)
## 
## Call:
## cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe +      Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem) 
## 
## Partitioning of scaled Chi-square:
##               Inertia Proportion
## Total          2.0832      1.000
## Constrained    1.4415      0.692
## Unconstrained  0.6417      0.308
## 
## Eigenvalues, and their contribution to the scaled Chi-square 
## 
## Importance of components:
##                         CCA1   CCA2    CCA3    CCA4    CCA5    CCA6    CCA7
## Eigenvalue            0.4389 0.2918 0.16285 0.14213 0.11795 0.08903 0.07029
## Proportion Explained  0.2107 0.1401 0.07817 0.06823 0.05662 0.04274 0.03374
## Cumulative Proportion 0.2107 0.3507 0.42890 0.49713 0.55375 0.59649 0.63023
##                          CCA8    CCA9    CCA10    CCA11    CCA12    CCA13
## Eigenvalue            0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained  0.02801 0.01495 0.006382 0.004015 0.003139 0.002955
## Cumulative Proportion 0.65825 0.67319 0.679576 0.683592 0.686730 0.689685
##                          CCA14     CA1     CA2     CA3     CA4     CA5     CA6
## Eigenvalue            0.004733 0.19776 0.14193 0.10117 0.07079 0.05330 0.03330
## Proportion Explained  0.002272 0.09493 0.06813 0.04857 0.03398 0.02559 0.01598
## Cumulative Proportion 0.691958 0.78689 0.85502 0.90359 0.93757 0.96315 0.97914
##                            CA7      CA8      CA9
## Eigenvalue            0.018868 0.015104 0.009488
## Proportion Explained  0.009057 0.007251 0.004554
## Cumulative Proportion 0.988195 0.995446 1.000000
## 
## Accumulated constrained eigenvalues
## Importance of components:
##                         CCA1   CCA2   CCA3   CCA4    CCA5    CCA6    CCA7
## Eigenvalue            0.4389 0.2918 0.1628 0.1421 0.11795 0.08903 0.07029
## Proportion Explained  0.3045 0.2024 0.1130 0.0986 0.08183 0.06176 0.04877
## Cumulative Proportion 0.3045 0.5069 0.6198 0.7184 0.80027 0.86203 0.91080
##                          CCA8    CCA9    CCA10    CCA11    CCA12    CCA13
## Eigenvalue            0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained  0.04049 0.02160 0.009223 0.005803 0.004536 0.004271
## Cumulative Proportion 0.95128 0.97288 0.982107 0.987910 0.992446 0.996716
##                          CCA14
## Eigenvalue            0.004733
## Proportion Explained  0.003284
## Cumulative Proportion 1.000000
## 
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
## 
## 
## Species scores
## 
##              CCA1     CCA2     CCA3      CCA4     CCA5     CCA6
## Callvulg  0.11374 -1.73246  4.15753  1.844837  3.13741 -1.15626
## Empenigr -0.27373  0.14089  0.09036 -1.134550 -0.40226  0.03525
## Rhodtome -1.59033 -0.11156  0.19187 -2.490433 -0.62292 -1.73616
## Vaccmyrt -1.92827  0.56943  0.75259 -0.244263 -1.65624 -2.05452
## Vaccviti -0.23029  0.22315 -0.13141 -0.960947  0.24441  0.02995
## Pinusylv  0.36674  0.48934  0.55326 -0.726273  0.85051 -0.21227
## Descflex -2.17952  0.50020 -0.40165  1.608949 -1.38617  1.28224
## Betupube -1.07326 -0.41989 -0.20569 -6.388345 -0.62954 -5.60316
## Vacculig  0.77560 -2.19992 -0.93608  0.469588 -2.78966  1.04277
## Diphcomp  0.14991 -1.65300 -1.03898 -1.412059 -0.78833  2.08551
## Dicrsp   -1.28302  0.42862 -4.34136  0.691801  4.43282  1.30777
## Dicrfusc -0.75393 -0.76901  2.04376 -0.684761  0.32655  2.14058
## Dicrpoly -0.79564  0.14903 -2.01239 -3.186680  2.23820 -3.43647
## Hylosple -2.75940  1.46965  0.12345  3.602353 -2.66866 -0.74851
## Pleuschr -1.39625  0.62360 -0.02266  0.817213 -0.19077  0.06281
## Polypili  0.21763 -0.84392 -1.27708 -0.747467 -0.15333  0.16978
## Polyjuni -0.91607  0.38917 -0.87255 -0.891253 -1.78446  1.17847
## Polycomm -1.34974  0.59359 -0.58214 -2.854381 -1.19037 -2.60320
## Pohlnuta -0.01435  0.46778 -0.34834 -0.931562  1.23465 -0.32446
## Ptilcili -0.86964 -0.22650 -0.14520 -5.594842 -0.48392 -5.05263
## Barbhatc -1.04773 -0.42524 -0.29330 -6.830157 -0.50320 -6.88497
## Cladarbu  0.31928 -1.31813 -0.06534  0.138503 -0.11811 -0.26229
## Cladrang  0.57516 -1.14185 -0.60438  0.280955 -0.47617  0.10938
## Cladstel  1.36834  1.29986  0.20555  0.179762 -0.04827  0.09185
## Cladunci -0.34820  0.11797 -0.03422 -1.037583  2.65119 -0.48962
## Cladcocc  0.33121 -0.25213  0.31806 -0.205437  0.09828  0.41903
## Cladcorn -0.34025  0.12974 -0.22432 -0.686052 -0.31883  0.57211
## Cladgrac -0.16429 -0.34432 -0.39566 -0.533216  0.70217 -0.07237
## Cladfimb  0.03022 -0.16993  0.47734 -0.696051 -0.10470 -0.11656
## Cladcris -0.20689  0.02979  1.04812 -1.124294  0.40186 -0.43505
## Cladchlo  0.66964  1.02386 -0.68975 -1.528620  0.49217 -0.75368
## Cladbotr -1.02718 -0.35199  0.48348 -3.528217  0.63524 -4.23041
## Cladamau -0.02415 -2.15363 -1.80591 -1.323302 -1.02050  2.39498
## Cladsp    1.03577  0.72454  0.76099  0.741439  1.75911  0.41843
## Cetreric  0.09754 -0.07199 -1.05941  0.315233  2.75328 -0.58261
## Cetrisla  0.24027  0.64936 -0.12182 -2.346145  0.48511 -2.31098
## Flavniva  1.31684 -1.19677 -1.15320  5.202080  1.07346 -7.81575
## Nepharct -1.15139  0.36798 -1.38414 -0.153781 -3.31081  2.49381
## Stersp    0.18370 -2.37391 -2.38790 -0.009846 -1.07525  1.39790
## Peltapht -0.60047  0.31181  0.12300 -0.899163 -0.76856  0.65021
## Icmaeric  0.26085 -2.83828 -1.06550 -0.409685 -1.20472  1.06913
## Cladcerv  1.06877 -0.10889 -0.78377  3.250753  0.01418 -3.50019
## Claddefo -0.45498 -0.03870  0.60324 -1.497542  0.85219 -0.63272
## Cladphyl  1.51291  2.08492  0.04117 -0.268421  0.27480  0.48796
## 
## 
## Site scores (weighted averages of species scores)
## 
##        CCA1     CCA2      CCA3     CCA4       CCA5     CCA6
## 18  0.11823 -0.57251 -0.164982 -0.22892 -0.1940177  0.07213
## 15 -0.64276 -0.10649  0.169910  0.11432  0.0521025  0.23988
## 24 -0.84786  0.25736 -1.189184  0.14813  1.3580797  0.22853
## 27 -0.99432  0.35227  0.034639  0.28730 -0.4232940 -0.02911
## 23 -0.39622 -0.09941 -0.054725 -0.43892 -0.1038877  0.02098
## 19 -0.07306  0.38585  0.006695 -0.02930 -0.1896126 -0.02464
## 22 -0.72351 -0.26482  0.855780 -0.16216  0.0893297  0.55882
## 16 -0.50071 -0.42518  0.666714 -0.05991  0.1632133  0.51821
## 28 -1.48534  0.62159  0.100450  0.70953 -0.6209907 -0.35786
## 13  0.26728 -0.79352  0.904036  0.45978  0.6372512 -0.27314
## 14 -0.30232 -0.37451  0.439688 -0.39404  0.9278442  0.04663
## 20 -0.36984 -0.13664 -0.135727 -0.13735  0.0942856  0.03259
## 25 -0.85602  0.13551 -0.587776 -0.01017  0.3304823  0.65496
## 7   0.36936 -1.08951 -0.372699  0.05638 -0.4616061  0.05740
## 5   0.44060 -1.21454 -0.658393  0.16630 -0.4226872  0.15976
## 6   0.39218 -0.69770 -0.189710 -0.03141 -0.0990151 -0.05450
## 3   0.88634  0.21282 -0.085773  0.09809 -0.2111364  0.08974
## 4   0.77344 -0.30247 -0.083929  0.80863  0.1228687 -0.94716
## 2   0.93351  0.60859  0.004559  0.01574 -0.1379707  0.08149
## 9   0.86982  0.91296  0.096369 -0.05063  0.0005494  0.01469
## 12  0.67010  0.58561  0.034417 -0.09231 -0.0424654  0.05488
## 10  0.93439  0.83587  0.093851 -0.06296 -0.0540425  0.05003
## 11  0.30814  0.02923 -0.059108  0.09765 -0.0281526 -0.01160
## 21 -0.47641  0.23201  0.003915 -1.44448 -0.2880137 -1.21174
## 
## 
## Site constraints (linear combinations of constraining variables)
## 
##        CCA1     CCA2      CCA3     CCA4      CCA5     CCA6
## 18 -0.28028 -0.71553 -0.198602 -0.35622 -0.016645  0.28042
## 15 -0.12605  0.26839  0.183427 -0.19962 -0.026309 -0.23569
## 24 -0.57190  0.13619 -1.113919  0.21486  1.130853  0.07844
## 27 -1.12491  0.26289 -0.227400  0.40474 -0.210924  0.14882
## 23 -0.52704  0.05792  0.103917 -0.34088 -0.098759  0.13090
## 19 -0.44851  0.54086  0.013494 -0.37832 -0.048521 -0.27997
## 22 -0.54244 -0.36270  0.612068 -0.02209  0.194629  0.66118
## 16 -0.09855 -0.62779  0.413117 -0.16871 -0.053130 -0.07504
## 28 -1.37258  0.59298  0.200796  0.71143 -0.478758 -0.19023
## 13  0.10953 -0.73196  1.049988  0.47179  0.604495 -0.16296
## 14 -0.09320  0.10867  0.313802 -0.33147  0.232443 -0.11453
## 20 -0.45423  0.04379 -0.082409 -0.42046  0.381970 -0.22780
## 25 -0.59995  0.15944 -0.222688 -0.02782 -0.388701  0.24252
## 7   0.91721 -1.04185 -0.323016  0.13738 -0.567910 -0.03541
## 5   0.06432 -1.09164 -0.636766  0.01507 -0.151542  0.29546
## 6   0.27735 -0.30740 -0.130894  0.02489 -0.019959  0.10057
## 3   0.63365  0.06729 -0.206032  0.05714 -0.365798 -0.04820
## 4   0.56735 -0.42871 -0.189592  0.87651  0.160886 -0.84789
## 2   1.01789  0.50232  0.038999  0.09780 -0.003433  0.21274
## 9   1.01611  0.86649 -0.006136 -0.04395  0.239962  0.19821
## 12  0.29646  0.12958  0.378874 -0.10628  0.044242  0.11423
## 10  0.73605  0.86077 -0.016804  0.04149 -0.158371  0.07950
## 11  0.39119  0.19766 -0.018368 -0.05333 -0.024357 -0.11579
## 21 -0.45499 -0.12585 -0.070007 -1.04926 -0.070611 -0.65096
## 
## 
## Biplot scores for constraining variables
## 
##              CCA1     CCA2      CCA3      CCA4      CCA5       CCA6
## N        -0.14766 -0.28570  0.002715  0.066860 -0.086965  0.0304388
## P        -0.21110  0.31268 -0.065374  0.180759  0.063227 -0.0363527
## K        -0.24254  0.16634  0.145204  0.180743  0.111771 -0.0586722
## Ca       -0.29655  0.22782 -0.015240  0.037048  0.105769  0.0129929
## Mg       -0.28817  0.18393 -0.057371  0.040679  0.170979 -0.0017181
## S        -0.01594  0.22454  0.059879  0.167562  0.205056 -0.0496189
## Al        0.50996 -0.02564  0.015177  0.147400  0.055261 -0.1004202
## Fe        0.43000 -0.04760 -0.016976  0.099139 -0.023974 -0.0332227
## Mn       -0.47852  0.12132  0.045621  0.109904 -0.047628  0.0538484
## Zn       -0.23723  0.18092 -0.112151  0.130337  0.212656 -0.0003564
## Mo        0.13523 -0.05582 -0.063359  0.122240  0.177367 -0.0935487
## Baresoil -0.35558 -0.13762  0.055249 -0.196250  0.057225 -0.1051508
## Humdepth -0.46157  0.10891  0.109612 -0.051173 -0.001117 -0.0153218
## pH        0.32935  0.04056 -0.131693  0.007887 -0.049995 -0.0176316
summary(vare.cca,scaling=2)
## 
## Call:
## cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe +      Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem) 
## 
## Partitioning of scaled Chi-square:
##               Inertia Proportion
## Total          2.0832      1.000
## Constrained    1.4415      0.692
## Unconstrained  0.6417      0.308
## 
## Eigenvalues, and their contribution to the scaled Chi-square 
## 
## Importance of components:
##                         CCA1   CCA2    CCA3    CCA4    CCA5    CCA6    CCA7
## Eigenvalue            0.4389 0.2918 0.16285 0.14213 0.11795 0.08903 0.07029
## Proportion Explained  0.2107 0.1401 0.07817 0.06823 0.05662 0.04274 0.03374
## Cumulative Proportion 0.2107 0.3507 0.42890 0.49713 0.55375 0.59649 0.63023
##                          CCA8    CCA9    CCA10    CCA11    CCA12    CCA13
## Eigenvalue            0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained  0.02801 0.01495 0.006382 0.004015 0.003139 0.002955
## Cumulative Proportion 0.65825 0.67319 0.679576 0.683592 0.686730 0.689685
##                          CCA14     CA1     CA2     CA3     CA4     CA5     CA6
## Eigenvalue            0.004733 0.19776 0.14193 0.10117 0.07079 0.05330 0.03330
## Proportion Explained  0.002272 0.09493 0.06813 0.04857 0.03398 0.02559 0.01598
## Cumulative Proportion 0.691958 0.78689 0.85502 0.90359 0.93757 0.96315 0.97914
##                            CA7      CA8      CA9
## Eigenvalue            0.018868 0.015104 0.009488
## Proportion Explained  0.009057 0.007251 0.004554
## Cumulative Proportion 0.988195 0.995446 1.000000
## 
## Accumulated constrained eigenvalues
## Importance of components:
##                         CCA1   CCA2   CCA3   CCA4    CCA5    CCA6    CCA7
## Eigenvalue            0.4389 0.2918 0.1628 0.1421 0.11795 0.08903 0.07029
## Proportion Explained  0.3045 0.2024 0.1130 0.0986 0.08183 0.06176 0.04877
## Cumulative Proportion 0.3045 0.5069 0.6198 0.7184 0.80027 0.86203 0.91080
##                          CCA8    CCA9    CCA10    CCA11    CCA12    CCA13
## Eigenvalue            0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained  0.04049 0.02160 0.009223 0.005803 0.004536 0.004271
## Cumulative Proportion 0.95128 0.97288 0.982107 0.987910 0.992446 0.996716
##                          CCA14
## Eigenvalue            0.004733
## Proportion Explained  0.003284
## Cumulative Proportion 1.000000
## 
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## 
## 
## Species scores
## 
##               CCA1     CCA2      CCA3      CCA4      CCA5      CCA6
## Callvulg  0.075347 -0.93581  1.677742  0.695507  1.077518 -0.345001
## Empenigr -0.181340  0.07610  0.036462 -0.427727 -0.138153  0.010517
## Rhodtome -1.053549 -0.06026  0.077428 -0.938897 -0.213938 -0.518031
## Vaccmyrt -1.277428  0.30759  0.303704 -0.092088 -0.568820 -0.613023
## Vaccviti -0.152563  0.12054 -0.053031 -0.362279  0.083942  0.008938
## Pinusylv  0.242956  0.26432  0.223265 -0.273806  0.292102 -0.063335
## Descflex -1.443872  0.27019 -0.162082  0.606576 -0.476067  0.382590
## Betupube -0.711004 -0.22681 -0.083007 -2.408417 -0.216212 -1.671857
## Vacculig  0.513817 -1.18831 -0.377748  0.177035 -0.958084  0.311138
## Diphcomp  0.099310 -0.89289 -0.419273 -0.532348 -0.270745  0.622270
## Dicrsp   -0.849964  0.23153 -1.751924  0.260810  1.522412  0.390210
## Dicrfusc -0.499460 -0.41539  0.824743 -0.258156  0.112149  0.638702
## Dicrpoly -0.527090  0.08050 -0.812083 -1.201383  0.768689 -1.025365
## Hylosple -1.828026  0.79385  0.049816  1.358093 -0.916528 -0.223338
## Pleuschr -0.924978  0.33684 -0.009146  0.308091 -0.065518  0.018741
## Polypili  0.144172 -0.45586 -0.515356 -0.281796 -0.052660  0.050659
## Polyjuni -0.606869  0.21021 -0.352109 -0.336004 -0.612858  0.351629
## Polycomm -0.894165  0.32063 -0.234919 -1.076106 -0.408823 -0.776736
## Pohlnuta -0.009508  0.25268 -0.140571 -0.351201  0.424031 -0.096811
## Ptilcili -0.576115 -0.12234 -0.058593 -2.109265 -0.166198 -1.507591
## Barbhatc -0.694092 -0.22970 -0.118360 -2.574980 -0.172821 -2.054320
## Cladarbu  0.211517 -0.71201 -0.026366  0.052216 -0.040564 -0.078262
## Cladrang  0.381030 -0.61678 -0.243893  0.105921 -0.163536  0.032637
## Cladstel  0.906486  0.70213  0.082949  0.067771 -0.016579  0.027407
## Cladunci -0.230671  0.06372 -0.013810 -0.391170  0.910527 -0.146092
## Cladcocc  0.219419 -0.13619  0.128350 -0.077450  0.033754  0.125028
## Cladcorn -0.225404  0.07008 -0.090524 -0.258643 -0.109501  0.170706
## Cladgrac -0.108836 -0.18599 -0.159664 -0.201023  0.241156 -0.021594
## Cladfimb  0.020022 -0.09179  0.192626 -0.262413 -0.035959 -0.034780
## Cladcris -0.137056  0.01609  0.422960 -0.423861  0.138016 -0.129810
## Cladchlo  0.443621  0.55305 -0.278345 -0.576292  0.169030 -0.224882
## Cladbotr -0.680481 -0.19013  0.195105 -1.330144  0.218169 -1.262258
## Cladamau -0.015996 -1.16331 -0.728763 -0.498887 -0.350481  0.714608
## Cladsp    0.686166  0.39137  0.307091  0.279524  0.604150  0.124850
## Cetreric  0.064619 -0.03889 -0.427516  0.118844  0.945590 -0.173838
## Cetrisla  0.159171  0.35076 -0.049161 -0.884501  0.166607 -0.689545
## Flavniva  0.872373 -0.64645 -0.465365  1.961193  0.368671 -2.332045
## Nepharct -0.762768  0.19877 -0.558560 -0.057976 -1.137069  0.744096
## Stersp    0.121697 -1.28229 -0.963619 -0.003712 -0.369284  0.417103
## Peltapht -0.397796  0.16843  0.049634 -0.338986 -0.263955  0.194009
## Icmaeric  0.172805 -1.53313 -0.429975 -0.154452 -0.413750  0.319003
## Cladcerv  0.708032 -0.05882 -0.316283  1.225539  0.004871 -1.044377
## Claddefo -0.301412 -0.02090  0.243431 -0.564576  0.292677 -0.188788
## Cladphyl  1.002262  1.12620  0.016613 -0.101195  0.094379  0.145598
## 
## 
## Site scores (weighted averages of species scores)
## 
##       CCA1     CCA2      CCA3     CCA4     CCA5     CCA6
## 18  0.1785 -1.05988 -0.408835 -0.60721 -0.56492  0.24175
## 15 -0.9702 -0.19714  0.421046  0.30324  0.15171  0.80394
## 24 -1.2798  0.47645 -2.946863  0.39292  3.95433  0.76592
## 27 -1.5009  0.65216  0.085837  0.76207 -1.23251 -0.09756
## 23 -0.5981 -0.18404 -0.135611 -1.16425 -0.30249  0.07033
## 19 -0.1103  0.71431  0.016591 -0.07773 -0.55210 -0.08258
## 22 -1.0921 -0.49026  2.120668 -0.43014  0.26010  1.87287
## 16 -0.7558 -0.78712  1.652152 -0.15892  0.47523  1.73677
## 28 -2.2421  1.15075  0.248921  1.88204 -1.80814 -1.19935
## 13  0.4035 -1.46904  2.240249  1.21956  1.85549 -0.91541
## 14 -0.4563 -0.69333  1.089571 -1.04519  2.70161  0.15628
## 20 -0.5583 -0.25296 -0.336340 -0.36433  0.27453  0.10923
## 25 -1.2922  0.25087 -1.456542 -0.02698  0.96227  2.19508
## 7   0.5576 -2.01700 -0.923568  0.14954 -1.34406  0.19237
## 5   0.6651 -2.24847 -1.631533  0.44110 -1.23074  0.53544
## 6   0.5920 -1.29165 -0.470112 -0.08331 -0.28830 -0.18265
## 3   1.3379  0.39399 -0.212551  0.26020 -0.61477  0.30075
## 4   1.1675 -0.55997 -0.207980  2.14490  0.35776 -3.17436
## 2   1.4091  1.12669  0.011297  0.04175 -0.40173  0.27311
## 9   1.3130  1.69016  0.238808 -0.13429  0.00160  0.04923
## 12  1.0115  1.08413  0.085287 -0.24485 -0.12365  0.18392
## 10  1.4105  1.54744  0.232569 -0.16699 -0.15736  0.16768
## 11  0.4651  0.05411 -0.146473  0.25902 -0.08197 -0.03886
## 21 -0.7191  0.42952  0.009702 -3.83149 -0.83861 -4.06109
## 
## 
## Site constraints (linear combinations of constraining variables)
## 
##        CCA1     CCA2     CCA3     CCA4      CCA5    CCA6
## 18 -0.42308 -1.32466 -0.49215 -0.94489 -0.048464  0.9398
## 15 -0.19026  0.49687  0.45454 -0.52951 -0.076603 -0.7899
## 24 -0.86328  0.25213 -2.76035  0.56993  3.292710  0.2629
## 27 -1.69805  0.48669 -0.56351  1.07358 -0.614147  0.4988
## 23 -0.79557  0.10723  0.25751 -0.90419 -0.287557  0.4387
## 19 -0.67702  1.00130  0.03344 -1.00351 -0.141279 -0.9383
## 22 -0.81881 -0.67147  1.51674 -0.05858  0.566703  2.2159
## 16 -0.14877 -1.16222  1.02373 -0.44751 -0.154699 -0.2515
## 28 -2.07190  1.09778  0.49758  1.88707 -1.394002 -0.6375
## 13  0.16534 -1.35508  2.60193  1.25142  1.760111 -0.5461
## 14 -0.14069  0.20118  0.77762 -0.87922  0.676806 -0.3838
## 20 -0.68566  0.08107 -0.20421 -1.11529  1.112185 -0.7635
## 25 -0.90562  0.29517 -0.55183 -0.07379 -1.131782  0.8128
## 7   1.38453 -1.92877 -0.80045  0.36440 -1.653585 -0.1187
## 5   0.09709 -2.02095 -1.57794  0.03999 -0.441247  0.9902
## 6   0.41866 -0.56908 -0.32436  0.06603 -0.058116  0.3371
## 3   0.95649  0.12458 -0.51056  0.15157 -1.065096 -0.1616
## 4   0.85641 -0.79366 -0.46982  2.32495  0.468453 -2.8417
## 2   1.53650  0.92994  0.09664  0.25941 -0.009995  0.7130
## 9   1.53381  1.60412 -0.01520 -0.11658  0.698700  0.6643
## 12  0.44751  0.23990  0.93887 -0.28191  0.128819  0.3828
## 10  1.11107  1.59354 -0.04164  0.11005 -0.461130  0.2664
## 11  0.59050  0.36592 -0.04552 -0.14145 -0.070919 -0.3881
## 21 -0.68681 -0.23299 -0.17348 -2.78317 -0.205599 -2.1817
## 
## 
## Biplot scores for constraining variables
## 
##              CCA1     CCA2      CCA3     CCA4      CCA5      CCA6
## N        -0.22290 -0.52891  0.006729  0.17735 -0.253216  0.102014
## P        -0.31866  0.57886 -0.162001  0.47947  0.184099 -0.121835
## K        -0.36612  0.30794  0.359824  0.47942  0.325444 -0.196637
## Ca       -0.44764  0.42176 -0.037765  0.09827  0.307969  0.043545
## Mg       -0.43499  0.34051 -0.142169  0.10790  0.497841 -0.005758
## S        -0.02406  0.41570  0.148384  0.44446  0.597063 -0.166296
## Al        0.76978 -0.04747  0.037610  0.39098  0.160905 -0.336554
## Fe        0.64909 -0.08811 -0.042067  0.26297 -0.069806 -0.111345
## Mn       -0.72232  0.22460  0.113052  0.29152 -0.138680  0.180471
## Zn       -0.35810  0.33493 -0.277916  0.34572  0.619191 -0.001195
## Mo        0.20413 -0.10334 -0.157007  0.32424  0.516439 -0.313525
## Baresoil -0.53675 -0.25477  0.136910 -0.52055  0.166621 -0.352409
## Humdepth -0.69673  0.20163  0.271625 -0.13574 -0.003252 -0.051350
## pH        0.49716  0.07509 -0.326341  0.02092 -0.145569 -0.059091
# again two different types of scaling are possible for triplots

# hypothesis tests #
# testing the first axis (global test)
anova(vare.cca)
## Permutation test for cca under reduced model
## Permutation: free
## Number of permutations: 999
## 
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##          Df ChiSquare      F Pr(>F)  
## Model    14   1.44148 1.4441  0.032 *
## Residual  9   0.64171                
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(vare.cca,first=TRUE)
## Permutation test for cca under reduced model
## Permutation: free
## Number of permutations: 999
## 
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##          Df ChiSquare      F Pr(>F)  
## CCA1      1   0.43887 6.1551  0.041 *
## Residual  9   0.64171                
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# testing all axes sequentially (preceding axes are taken as constraints)
anova(vare.cca,by="axis",model="direct",perm.max=9999,step=1000)
## Permutation test for cca under direct model
## Forward tests for axes
## Permutation: free
## Number of permutations: 999
## 
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##          Df ChiSquare      F Pr(>F)  
## CCA1      1   0.43887 6.1551  0.029 *
## CCA2      1   0.29178 4.0921  0.401  
## CCA3      1   0.16285 2.2839  0.962  
## CCA4      1   0.14213 1.9934  0.970  
## CCA5      1   0.11795 1.6543  0.994  
## CCA6      1   0.08903 1.2486  1.000  
## CCA7      1   0.07029 0.9859  1.000  
## CCA8      1   0.05836 0.8185  1.000  
## CCA9      1   0.03114 0.4367  1.000  
## CCA10     1   0.01329 0.1865  1.000  
## CCA11     1   0.00836 0.1173  1.000  
## CCA12     1   0.00654 0.0917  1.000  
## CCA13     1   0.00616 0.0863  1.000  
## CCA14     1   0.00473 0.0664  1.000  
## Residual  9   0.64171                
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# testing the individual terms=constraints
anova(vare.cca,by="terms",model="direct",perm.max=9999,step=1000)  # tests terms sequentially, order matters!
## Permutation test for cca under direct model
## Terms added sequentially (first to last)
## Permutation: free
## Number of permutations: 999
## 
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##          Df ChiSquare      F Pr(>F)  
## N         1   0.13001 1.8234  0.053 .
## P         1   0.17996 2.5240  0.018 *
## K         1   0.13643 1.9134  0.065 .
## Ca        1   0.08614 1.2081  0.279  
## Mg        1   0.06760 0.9481  0.460  
## S         1   0.17545 2.4606  0.012 *
## Al        1   0.14797 2.0753  0.035 *
## Fe        1   0.04982 0.6987  0.723  
## Mn        1   0.05080 0.7124  0.748  
## Zn        1   0.07167 1.0052  0.452  
## Mo        1   0.09585 1.3442  0.219  
## Baresoil  1   0.08683 1.2178  0.267  
## Humdepth  1   0.09502 1.3327  0.228  
## pH        1   0.06794 0.9529  0.472  
## Residual  9   0.64171                
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(vare.cca,by="margin",model="direct",perm.max=9999,step=1000) # tests each term in full model (like drop1() function)
## Permutation test for cca under direct model
## Marginal effects of terms
## Permutation: free
## Number of permutations: 999
## 
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##          Df ChiSquare      F Pr(>F)  
## N         1   0.06302 0.8838  0.548  
## P         1   0.10327 1.4484  0.177  
## K         1   0.10385 1.4565  0.140  
## Ca        1   0.06169 0.8652  0.589  
## Mg        1   0.08830 1.2384  0.258  
## S         1   0.10388 1.4569  0.166  
## Al        1   0.06876 0.9644  0.499  
## Fe        1   0.04765 0.6683  0.748  
## Mn        1   0.08110 1.1375  0.304  
## Zn        1   0.06448 0.9043  0.512  
## Mo        1   0.07761 1.0885  0.351  
## Baresoil  1   0.08966 1.2574  0.272  
## Humdepth  1   0.13548 1.9001  0.050 *
## pH        1   0.06794 0.9529  0.490  
## Residual  9   0.64171                
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# quite a lot of variables in the constraining matrix, maybe selection would be adequate
# --> function ordistep()

###################
# making triplots #

# again various types of scaling for the plotting step:
# scaling 1 "distance triplot"
# sites can be projected on constraints
# sites close to centroid of factor constraint are more likely to possess the specific state (factor level)
# distances among sites reflect their Chi^2 distances

# scaling 2 "correlation triplot"
# species can be projected on constraints (to give their optimum)
# species close to centroid of factor constraint are more likely to be found in the respective sites
# distances among sites do not reflect their Chi^2 distances

# scaling 3 is compromise

plot(vare.cca,scaling=1,display=c("species","sites"))
plot(vare.cca,scaling=1,display=c("species","sites","bp"))

plot(vare.cca,scaling=2)

plot(vare.cca,scaling=3) # a compromise

# for any scaling take care when interpreting species close to origin:
# these are "everywhere" or have optimum right at the origin (i.e., optimum with regard to both axis shown)

# for more controlled plotting compute scores individually

(species<-scores(vare.cca,display="species",scaling=2))
##                  CCA1        CCA2
## Callvulg  0.075346828 -0.93580864
## Empenigr -0.181339634  0.07610159
## Rhodtome -1.053549303 -0.06026078
## Vaccmyrt -1.277428382  0.30758571
## Vaccviti -0.152563159  0.12053851
## Pinusylv  0.242955730  0.26432438
## Descflex -1.443872268  0.27018775
## Betupube -0.711004196 -0.22680891
## Vacculig  0.513817276 -1.18831302
## Diphcomp  0.099309627 -0.89289046
## Dicrsp   -0.849963650  0.23152562
## Dicrfusc -0.499459808 -0.41538963
## Dicrpoly -0.527089621  0.08050066
## Hylosple -1.828025570  0.79385044
## Pleuschr -0.924977860  0.33684271
## Polypili  0.144171903 -0.45585592
## Polyjuni -0.606869478  0.21021273
## Polycomm -0.894165298  0.32063347
## Pohlnuta -0.009508326  0.25267888
## Ptilcili -0.576115080 -0.12234488
## Barbhatc -0.694091803 -0.22969682
## Cladarbu  0.211516504 -0.71200604
## Cladrang  0.381029817 -0.61678185
## Cladstel  0.906485693  0.70213449
## Cladunci -0.230670950  0.06372124
## Cladcocc  0.219418695 -0.13619040
## Cladcorn -0.225403730  0.07008025
## Cladgrac -0.108836318 -0.18598627
## Cladfimb  0.020022402 -0.09178764
## Cladcris -0.137056217  0.01608948
## Cladchlo  0.443621232  0.55305019
## Cladbotr -0.680480562 -0.19013376
## Cladamau -0.015995645 -1.16331056
## Cladsp    0.686166357  0.39136838
## Cetreric  0.064619228 -0.03888872
## Cetrisla  0.159171117  0.35076227
## Flavniva  0.872372656 -0.64644751
## Nepharct -0.762767923  0.19876924
## Stersp    0.121696721 -1.28229477
## Peltapht -0.397796035  0.16842583
## Icmaeric  0.172805490 -1.53313439
## Cladcerv  0.708032101 -0.05881737
## Claddefo -0.301411810 -0.02090192
## Cladphyl  1.002261821  1.12619548

(lcs<-scores(vare.cca,display="lc",scaling=2)) # fitted site scores
##           CCA1        CCA2
## 18 -0.42308410 -1.32465581
## 15 -0.19026462  0.49687345
## 24 -0.86328447  0.25212611
## 27 -1.69805066  0.48669129
## 23 -0.79556975  0.10723446
## 19 -0.67702233  1.00129602
## 22 -0.81881442 -0.67146524
## 16 -0.14876518 -1.16222054
## 28 -2.07190438  1.09778134
## 13  0.16533766 -1.35507570
## 14 -0.14068713  0.20117513
## 20 -0.68566132  0.08107181
## 25 -0.90561895  0.29516717
## 7   1.38453057 -1.92876582
## 5   0.09708596 -2.02094655
## 6   0.41865544 -0.56908384
## 3   0.95649196  0.12457945
## 4   0.85641341 -0.79366326
## 2   1.53649728  0.92993610
## 9   1.53380971  1.60412304
## 12  0.44750585  0.23989837
## 10  1.11106621  1.59354430
## 11  0.59049933  0.36592318
## 21 -0.68681020 -0.23299449

(sites<-scores(vare.cca,display="sites",scaling=2)) # unfitted site scores
##          CCA1        CCA2
## 18  0.1784733 -1.05988423
## 15 -0.9702382 -0.19713866
## 24 -1.2798478  0.47644978
## 27 -1.5009195  0.65215591
## 23 -0.5980933 -0.18403623
## 19 -0.1102881  0.71431421
## 22 -1.0921288 -0.49026245
## 16 -0.7558244 -0.78712482
## 28 -2.2421137  1.15075172
## 13  0.4034539 -1.46904155
## 14 -0.4563466 -0.69332916
## 20 -0.5582740 -0.25295922
## 25 -1.2921584  0.25086578
## 7   0.5575516 -2.01700439
## 5   0.6650822 -2.24846553
## 6   0.5919915 -1.29165187
## 3   1.3379214  0.39399435
## 4   1.1674982 -0.55996655
## 2   1.4091299  1.12668910
## 9   1.3129824  1.69015916
## 12  1.0115068  1.08413112
## 10  1.4104517  1.54743675
## 11  0.4651334  0.05410696
## 21 -0.7191331  0.42951771

(constraints<-scores(vare.cca,choices=c(1,2),display="bp",scaling=2))
##                 CCA1        CCA2
## N        -0.22289549 -0.52891359
## P        -0.31865680  0.57886019
## K        -0.36611630  0.30794063
## Ca       -0.44763820  0.42176452
## Mg       -0.43499067  0.34051494
## S        -0.02405594  0.41569739
## Al        0.76977830 -0.04747443
## Fe        0.64908911 -0.08811470
## Mn       -0.72232495  0.22459941
## Zn       -0.35809796  0.33492792
## Mo        0.20413234 -0.10334347
## Baresoil -0.53674724 -0.25477013
## Humdepth -0.69673051  0.20163085
## pH        0.49715734  0.07508792

plot(sites,col="black",pch=21,xlim=c(-2,2),ylim=c(-2,2))
text(species,col="red",label=names(varespec),cex=0.7)
Arrows(x0=0,y0=0,x1=constraints[,1],y1=constraints[,2],lwd=1.5,col="blue")
text(constraints[,1:2]*1.1,label=rownames(constraints),pos=4,cex=0.8,col="blue")